Friedrich, Tobias; Kötzing, Timo; Radhakrishnan, Aishwarya; Schiller, Leon; Schirneck, Martin; Tennigkeit, Georg; Wietheger, Simon Crossover for Cardinality Constrained OptimizationACM Transactions on Evolutionary Learning and Optimization 2023: 1–32
To understand better how and why crossover can benefit constrained optimization, we consider pseudo-Boolean functions with an upper bound \(B\) on the number of 1-bits allowed in the length-\(n\) bit string (i.e., a cardinality constraint). We investigate the natural translation of the OneMax test function to this setting, a linear function where \(B\) bits have a weight of \(1+ 1/n\) and the remaining bits have a weight of \(1\). Friedrich et al. [TCS 2020] gave a bound of \(\Theta(n^2)\) for the expected running time of the (1+1) EA on this function. Part of the difficulty when optimizing this problem lies in having to improve individuals meeting the cardinality constraint by flipping a \(1\) and a \(0\) simultaneously. The experimental literature proposes balanced operators, preserving the number of 1-bits, as a remedy. We show that a balanced mutation operator optimizes the problem in \(O(n \log n)\) if \(n-B = O(1)\). However, if \(n-B = \Theta(n)\), we show a bound of \(\Omega(n^2)\), just as for classic bit mutation. Crossover together with a simple island model gives running times of \(O(n^2 / \log n)\) (uniform crossover) and \(O(n\sqrt{n})\) (3-ary majority vote crossover). For balanced uniform crossover with Hamming-distance maximization for diversity we show a bound of \(O(n \log n)\). As an additional contribution, we present an extensive analysis of different balanced crossover operators from the literature.
Friedrich, Tobias; Gawendowicz, Hans; Lenzner, Pascal; Melnichenko, Anna Social Distancing Network CreationAlgorithmica 2023
During a pandemic people have to find a trade-off between meeting others and staying safely at home. While meeting others is pleasant, it also increases the risk of infection. We consider this dilemma by introducing a game-theoretic network creation model in which selfish agents can form bilateral connections. They benefit from network neighbors, but at the same time, they want to maximize their distance to all other agents. This models the inherent conflict that social distancing rules impose on the behavior of selfish agents in a social network. Besides addressing this familiar issue, our model can be seen as the inverse to the well-studied Network Creation Game by Fabrikant et al. (in: PODC 2003, pp 347–351, 2003. https://doi.org/10.1145/872035.872088), where agents aim at being as central as possible in the created network. We look at two variants of network creation governed by social distancing. Firstly, a variant without connection restrictions, where we characterize optimal and equilibrium networks, and derive asymptotically tight bounds on the Price of Anarchy and Price of Stability. The second variant allows connection restrictions. As our main result, we prove that Swap-Maximal Routing-Cost Spanning Trees, an efficiently computable weaker variant of Maximum Routing-Cost Spanning Trees, actually resemble equilibria for a significant range of the parameter space. Moreover, we give almost tight bounds on the Price of Anarchy and Price of Stability. These results imply that under social distancing the agents’ selfishness has a strong impact on the quality of the equilibria.
Bläsius, Thomas; Friedrich, Tobias; Krejca, Martin S.; Molitor, Louise The Impact of Geometry on Monochrome Regions in the Flip Schelling ProcessComputational Geometry (CGTA) 2023: 101902
Schelling's classical segregation model gives a coherent explanation for the wide-spread phenomenon of residential segregation. We introduce an agent-based saturated open-city variant, the Flip Schelling Process (FSP), in which agents, placed on a graph, have one out of two types and, based on the predominant type in their neighborhood, decide whether to change their types; similar to a new agent arriving as soon as another agent leaves the vertex. We investigate the probability that an edge u,v is monochrome, i.e., that both vertices u and v have the same type in the FSP, and we provide a general framework for analyzing the influence of the underlying graph topology on residential segregation. In particular, for two adjacent vertices, we show that a highly decisive common neighborhood, i.e., a common neighborhood where the absolute value of the difference between the number of vertices with different types is high, supports segregation and, moreover, that large common neighborhoods are more decisive. As an application, we study the expected behavior of the FSP on two common random graph models with and without geometry: (1) For random geometric graphs, we show that the existence of an edge u,v makes a highly decisive common neighborhood for u and v more likely. Based on this, we prove the existence of a constant c>0 such that the expected fraction of monochrome edges after the FSP is at least 1/2+c. (2) For Erdős–Rényi graphs we show that large common neighborhoods are unlikely and that the expected fraction of monochrome edges after the FSP is at most 1/2+o(1). Our results indicate that the cluster structure of the underlying graph has a significant impact on the obtained segregation strength.
Paul, Subhabrata; Pradhan, Dinabandhu; Verma, Shaily Vertex-edge domination in interval and bipartite permutation graphsDiscuss. Math. Graph Theory 2023: 947–963
Given a graph \(G = (V,E)\), a vertex \(u \in V\) ve-dominates all edges incident to any vertex of \(N_G[u]\). A set \(D \subseteq V\) is a vertex-edge dominating set if for any edge \(e \in E\), there exists a vertex \(u \in D\) such that \(u\) ve-dominates \(e\). Given a graph \(G\), our goal is to find a minimum cardinality ve-dominating set of \(G\). In this paper, we designed two linear time algorithms to find a minimum cardinality ve-dominating set for interval and bipartite permutation graphs.
Böther, Maximilian; Schiller, Leon; Fischbeck, Philipp; Molitor, Louise; Krejca, Martin S.; Friedrich, Tobias Evolutionary Minimization of Traffic CongestionIEEE Transactions on Evolutionary Computation 2023: 1809–1821
Traffic congestion is a major issue that can be solved by suggesting drivers alternative routes they are willing to take. This concept has been formalized as a strategic routing problem in which a single alternative route is suggested to an existing one. We extend this formalization and introduce the Multiple-Routes problem, which is given a start and destination and aims at finding up to \(n\) different routes that the drivers strategically disperse over, minimizing the overall travel time of the system. Due to the NP-hard nature of the problem, we introduce the Multiple-Routes evolutionary algorithm (MREA) as a heuristic solver. We study several mutation and crossover operators and evaluate them on real-world data of Berlin, Germany. We find that a combination of all operators yields the best result, reducing the overall travel time by a factor between \(1.8\) and \(3\), in the median, compared to all drivers taking the fastest route. For the base case \(n = 2\), we compare our MREA to the highly tailored optimal solver by Bläsius et al. (ATMOS 2020), and show that, in the median, our approach finds solutions of quality at least \(99.69 \%\) of an optimal solution while only requiring \(40 \%\) of the time.
Behrendt, Lukas; Casel, Katrin; Friedrich, Tobias; Lagodzinski, J. A. Gregor; Löser, Alexander; Wilhelm, Marcus From symmetry to asymmetry: Generalizing TSP approximations by parametrizationJournal of Computer and System Sciences 2023: 157–170
We generalize the tree doubling and Christofides algorithm to parameterized approximations for ATSP (constant factor approximations that invest more runtime with respect to a chosen parameter). The parameters we consider are upper bounded by the number of asymmetric distances, which yields algorithms to efficiently compute good approximations for moderately asymmetric TSP instances. As generalization of the Christofides algorithm, we derive a parameterized 2.5-approximation, with the size of a vertex cover for the subgraph induced by the edges with asymmetric distances as parameter. Our generalization of tree doubling gives a parameterized 3-approximation, where the parameter is the minimum number of asymmetric distances in a minimum spanning arborescence. Further, we combine these with a notion of symmetry relaxation which allows to trade approximation guarantee for runtime. Since the parameters we consider are theoretically incomparable, we present experimental results showing that generalized tree doubling frequently outperforms generalized Christofides with respect to parameter size.
Krämer, Bastian; Stang, Moritz; Doskoč, Vanja; Schäfers, Wolfgang; Friedrich, Tobias Automated valuation models: improving model performance by choosing the optimal spatial training levelJournal of Property Research 2023: 365–390
The academic community has discussed using Automated Valuation Models (AVMs) in the context of traditional real estate valuations and their performance for several decades. Most studies focus on finding the best method for estimating property values. One aspect that has not yet to be studied scientifically is the appropriate choice of the spatial training level. The published research on AVMs usually deals with a manually defined region and fails to test the methods used on different spatial levels. Our research aims to investigate the impact of training AVM algorithms at different spatial levels regarding valuation accuracy. We use a dataset with 1.2 million residential properties from Germany and test four methods: Ordinary Least Square, Generalised Additive Models, eXtreme Gradient Boosting and Deep Neural Network. Our results show that the right choice of spatial training level can significantly impact the model performance, and that this impact varies across the different methods.
Cohen, Sarel; Hershcovitch, Moshik; Taraz, Martin; Kißig, Otto; Issac, Davis; Wood, Andrew; Waddington, Daniel; Chin, Peter; Friedrich, Tobias Improved And Optimized Drug Repurposing For The SARS-CoV-2 PandemicPlos One 2023
The active global SARS-CoV-2 pandemic caused more than 426 million cases and 5.8 million deaths worldwide. The development of completely new drugs for such a novel disease is a challenging, time intensive process. Despite researchers around the world working on this task, no effective treatments have been developed yet. This emphasizes the importance of drug repurposing, where treatments are found among existing drugs that are meant for different diseases. A common approach to this is based on knowledge graphs, that condense relationships between entities like drugs, diseases and genes. Graph neural networks (GNNs) can then be used for the task at hand by predicting links in such knowledge graphs. Expanding on state-of-the-art GNN research, Doshi et al. recently developed the Dr-COVID model. We further extend their work using additional output interpretation strategies. The best aggregation strategy derives a top-100 ranking of 8,070 candidate drugs, 32 of which are currently being tested in COVID-19-related clinical trials. Moreover, we present an alternative application for the model, the generation of additional candidates based on a given pre-selection of drug candidates using collaborative filtering. In addition, we improved the implementation of the Dr-COVID model by significantly shortening the inference and pre-processing time by exploiting data-parallelism. As drug repurposing is a task that requires high computation and memory resources, we further accelerate the post-processing phase using a new emerging hardware - we propose a new approach to leverage the use of high-capacity Non-Volatile Memory for aggregate drug ranking.
Friedrich, Tobias; Göbel, Andreas; Krejca, Martin S.; Pappik, Marcus Polymer Dynamics via Cliques: New Conditions for ApproximationsTheoretical Computer Science 2023: 230–252
Abstract polymer models are systems of weighted objects, called polymers, equipped with an incompatibility relation. An important quantity associated with such models is the partition function, which is the weighted sum over all sets of compatible polymers. Various approximation problems reduce to approximating the partition function of a polymer model. Central to the existence of such approximation algorithms are weight conditions of the respective polymer model. Such conditions are derived either via complex analysis or via probabilistic arguments. We follow the latter path and establish a new condition—the clique dynamics condition—, which is less restrictive than the ones in the literature. We introduce a new Markov chain where the clique dynamics condition implies rapid mixing by utilizing cliques of incompatible polymers that naturally arise from the translation of algorithmic problems into polymer models. This leads to improved parameter ranges for several approximation algorithms, such as a factor of at least \(2^{1/\alpha}\) for the hard-core model on bipartite \(\alpha\)-expanders.
Bläsius, Thomas; Fischbeck, Philipp; Friedrich, Tobias; Katzmann, Maximilian Solving Vertex Cover in Polynomial Time on Hyperbolic Random GraphsTheory of Computing Systems 2023: 28–51
The computational complexity of the VertexCover problem has been studied extensively. Most notably, it is NP-complete to find an optimal solution and typically NP-hard to find an approximation with reasonable factors. In contrast, recent experiments suggest that on many real-world networks the run time to solve VertexCover is way smaller than even the best known FPT-approaches can explain. We link these observations to two properties that are observed in many real-world networks, namely a heterogeneous degree distribution and high clustering. To formalize these properties and explain the observed behavior, we analyze how a branch-and-reduce algorithm performs on hyperbolic random graphs, which have become increasingly popular for modeling real-world networks. In fact, we are able to show that the VertexCover problem on hyperbolic random graphs can be solved in polynomial time, with high probability. The proof relies on interesting structural properties of hyperbolic random graphs. Since these predictions of the model are interesting in their own right, we conducted experiments on real-world networks showing that these properties are also observed in practice.
Wagner, Markus; Kohlros, Erik; Quantmeyer, Gerome; Kotzing, Timo ELEA – Build your own Evolutionary Algorithm in your BrowserGenetic and Evolutionary Computation Conference (GECCO ’23) 2023
We provide an open source framework to experiment with evolutionary algorithms which we call Experimenting and Learning toolkit for Evolutionary Algorithms (ELEA). ELEA is browser-based and allows to assemble evolutionary algorithms using drag-and-drop, starting from a number of simple predesigned examples, making the startup costs for employing the toolkit minimal. The designed examples can be executed and collected data can be displayed graphically. Further features include export of algorithm designs and experimental results as well as multi-threading. With the very intuitive user interface and the short time to get initial experiments going, this tool is especially suitable for explorative analyses of algorithms as well as for the use in classrooms.
Friedrich, Tobias; Lenzner, Pascal; Molitor, Louise; Seifert, Lars Single-Peaked Jump Schelling GamesInternational Symposium on Algorithmic Game Theory (SAGT) 2023
Schelling games model the wide-spread phenomenon of residential segregation in metropolitan areas from a game-theoretic point of view. In these games agents of different types each strategically select a node on a given graph that models the residential area to maximize their individual utility. The latter solely depends on the types of the agents on neighboring nodes and it has been a standard assumption to consider utility functions that are monotone in the number of same-type neighbors. This simplifying assumption has recently been challenged since sociological poll results suggest that real-world agents actually favor diverse neighborhoods. We contribute to the recent endeavor of investigating residential segregation models with realistic agent behavior by studying Jump Schelling Games with agents having a single-peaked utility function. In such games, there are empty nodes in the graph and agents can strategically jump to such nodes to improve their utility. We investigate the existence of equilibria and show that they exist under specific conditions. Contrasting this, we prove that even on simple topologies like paths or rings such stable states are not guaranteed to exist. Regarding the game dynamics, we show that improving response cycles exist independently of the position of the peak in the utility function. Moreover, we show high almost tight bounds on the Price of Anarchy and the Price of Stability with respect to the recently proposed degree of integration, which counts the number of agents with a diverse neighborhood and which serves as a proxy for measuring the segregation strength. Last but not least, we show that computing a beneficial state with high integration is NP-complete and, as a novel conceptual contribution, we also show that it is NP-hard to decide if an equilibrium state can be found via improving response dynamics starting from a given initial state.
Quinzan, Francesco; Khanna, Rajiv; Hershcovitch, Moshik; Cohen, Sarel; Waddington, Daniel G.; Friedrich, Tobias; Mahoney, Michael W. Fast Feature Selection with Fairness ConstraintsArtificial Intelligence and Statistics (AISTATS) 2023: 7800–7823
We study the fundamental problem of selecting optimal features for model construction. This problem is computationally challenging on large datasets, even with the use of greedy algorithm variants. To address this challenge, we extend the adaptive query model, recently proposed for the greedy forward selection for submodular functions, to the faster paradigm of Orthogonal Matching Pursuit for non-submodular functions. The proposed algorithm achieves exponentially fast parallel run time in the adaptive query model, scaling much better than prior work. Furthermore, our extension allows the use of downward-closed constraints, which can be used to encode certain fairness criteria into the feature selection process. We prove strong approximation guarantees for the algorithm based on standard assumptions. These guarantees are applicable to many parametric models, including Generalized Linear Models. Finally, we demonstrate empirically that the proposed algorithm competes favorably with state-of-the-art techniques for feature selection, on real-world and synthetic datasets.
Krogmann, Simon; Lenzner, Pascal; Skopalik, Alexander Strategic Facility Location with Clients that Minimize Total Waiting TimeConference on Artificial Intelligence (AAAI) 2023: 5714–5721
We study a non-cooperative two-sided facility location game in which facilities and clients behave strategically. This is in contrast to many other facility location games in which clients simply visit their closest facility. Facility agents select a location on a graph to open a facility to attract as much purchasing power as possible, while client agents choose which facilities to patronize by strategically distributing their purchasing power in order to minimize their total waiting time. Here, the waiting time of a facility depends on its received total purchasing power. We show that our client stage is an atomic splittable congestion game, which implies existence, uniqueness and efficient computation of a client equilibrium. Therefore, facility agents can efficiently predict client behavior and make strategic decisions accordingly. Despite that, we prove that subgame perfect equilibria do not exist in all instances of this game and that their existence is NP-hard to decide. On the positive side, we provide a simple and efficient algorithm to compute 3-approximate subgame perfect equilibria.
Blažej, Václav; Ganian, Robert; Knop, Dušan; Pokorný, Jan; Schierreich, Šimon; Simonov, Kirill The Parameterized Complexity of Network MicroaggregationConference on Artificial Intelligence (AAAI) 2023: 6262–6270
Microaggregation is a classical statistical disclosure control technique which requires the input data to be partitioned into clusters while adhering to specified size constraints. We provide novel exact algorithms and lower bounds for the task of microaggregating a given network while considering both unrestricted and connected clusterings, and analyze these from the perspective of the parameterized complexity paradigm. Altogether, our results assemble a complete complexity-theoretic picture for the network microaggregation problem with respect to the most natural parameteri- zations of the problem, including input-specified parameters capturing the size and homogeneity of the clusters as well as the treewidth and vertex cover number of the network.
Brand, Cornelius; Ganian, Robert; Simonov, Kirill A Parameterized Theory of PAC LearningConference on Artificial Intelligence (AAAI) 2023: 6834–6841
Probably Approximately Correct (i.e., PAC) learning is a core concept of sample complexity theory, and efficient PAC learnability is often seen as a natural counterpart to the class P in classical computational complexity. But while the nascent theory of parameterized complexity has allowed us to push beyond the P-NP “dichotomy” in classical computational complexity and identify the exact boundaries of tractability for numerous problems, there is no analogue in the domain of sample complexity that could push beyond efficient PAC learnability. As our core contribution, we fill this gap by developing a theory of parameterized PAC learning which allows us to shed new light on several recent PAC learning results that incorporated elements of parameterized complexity. Within the theory, we identify not one but two notions of fixed-parameter learnability that both form distinct counterparts to the class FPT—the core concept at the center of the parameterized complexity paradigm—and develop the machinery required to exclude fixed-parameter learnability. We then showcase the applications of this theory to identify refined boundaries of tractability for CNF and DNF learning as well as for a range of learning problems on graphs.
Cseh, Ágnes; Führlich, Pascal; Lenzner, Pascal The Swiss GambitAutonomous Agents and Multi-Agent Systems (AAMAS) 2023
In each round of a Swiss-system tournament, players of similar score are paired against each other. An intentional early loss therefore might lead to weaker opponents in later rounds and thus to a better final tournament result a phenomenon known as the Swiss Gambit. To the best of our knowledge it is an open question whether this strategy can actually work. This paper provides answers based on an empirical agent-based analysis for the most prominent application area of the Swiss-system format, namely chess tournaments. We simulate realistic tournaments by employing the official FIDE pairing system for computing the player pairings in each round. We show that even though gambits are widely possible in Swiss-system chess tournaments, profiting from them requires a high degree of predictability of match results. Moreover, even if a Swiss Gambit succeeds, the obtained improvement in the final ranking is limited. Our experiments prove that counting on a Swiss Gambit is indeed a lot more of a risky gambit than a reliable strategy to improve the final rank.
Bertschinger, Nils; Hoefer, Martin; Krogmann, Simon; Lenzner, Pascal; Schuldenzucker, Steffen; Wilhelmi, Lisa Equilibria and Convergence in Fire Sale GamesAutonomous Agents and Multiagent Systems (AAMAS) 2023: 215–223
The complex interactions between algorithmic trading agents can have a severe influence on the functioning of our economy, as witnessed by recent banking crises and trading anomalies. A common phenomenon in these situations are fire sales, a contagious process of asset sales that trigger further sales. We study the existence and structure of equilibria in a game-theoretic model of fire sales. We prove that for a wide parameter range (e.g., convex price impact functions), equilibria exist and form a complete lattice. This is contrasted with a non-existence result for concave price impact functions. Moreover, we study the convergence of best-response dynamics towards equilibria when they exist. In general, best-response dynamics may cycle. However, in many settings they are guaranteed to converge to the socially optimal equilibrium when starting from a natural initial state. Moreover, we discuss a simplified variant of the dynamics that is less informationally demanding and converges to the same equilibria. We compare the dynamics in terms of convergence speed.
Doering, Michelle; Peters, Jannik Margin of Victory for Weighted Tournament SolutionsAutonomous Agents and Multi-Agent Systems (AAMAS) 2023: 1716–1724
Determining how close a winner of an election is to becoming a loser, or distinguishing between different possible winners of an election, are major problems in computational social choice. We tackle these problems for so-called weighted tournament solutions by generalizing the notion of margin of victory (MoV) for tournament solutions by Brill et al. [2022][Artificial Intelligence] to weighted tournament solutions. For these, the MoV of a winner (resp. loser) is the total weight that needs to be changed in the tournament to make them a loser (resp. winner). We study three weighted tournament solutions: Borda’s rule, the weighted Uncovered Set, and Split Cycle. For all three rules, we determine whether the MoV for winners and non-winners is tractable and give upper and lower bounds on the possible values of the MoV. Further, we axiomatically study and generalize properties from the unweighted tournament setting to weighted tournaments.
Deligkas, Argyrios; Eiben, Eduard; Goldsmith, Tiger-Lily; Skretas, George Being an Influencer is Hard: The Complexity of Influence Maximization in Temporal Graphs with a Fixed SourceAutonomous Agents and Multi-Agent Systems (AAMAS) 2023: 2222–2230
We consider the influence maximization problem over a temporal graph, where there is a single fixed source. We deviate from the standard model of influence maximization, where the goal is to choose the set of most influential vertices. Instead, in our model we are given a fixed vertex, or source, and the goal is to find the best time steps to transmit so that the influence of this vertex is maximized. We frame this problem as a spreading process that follows a variant of the susceptible-infected-susceptible (SIS) model and we focus on three objective functions.In the MaxSpread objective, the goal is to maximize the total number of vertices that get infected at least once. In the MaxViral objective, the goal is to maximize the number of vertices that are infected at the same time step. Finally, in MaxViralTstep, the goal is to maximize the number of vertices that are infected at a given time step. We perform a thorough complexity theoretic analysis for these three objectives over three different scenarios: (1) the unconstrained setting where the source can transmit whenever it wants; (2) the window-constrained setting where the source has to transmit at either a predetermined, or a shifting window; (3) the periodic setting where the temporal graph has a small period. We prove that all of these problems, with the exception of MaxSpread for periodic graphs, are intractable even for very simple underlying graphs.
Friedrich, Tobias; Lenzner, Pascal; Molitor, Louise; Seifert, Lars Single-Peaked Jump Schelling GamesAutonomous Agents and Multiagent Systems (AAMAS) 2023: 2899–2901
Schelling games model the wide-spread phenomenon of residential segregation in metropolitan areas from a game-theoretic point of view. In these games agents of different types each strategically select a node on a given graph that models the residential area to maximize their individual utility. The latter solely depends on the types of the agents on neighboring nodes and it has been a standard assumption to consider utility functions that are monotone in the number of same-type neighbors. This simplifying assumption has recently been challenged since sociological poll results suggest that real-world agents actually favor diverse neighborhoods. We contribute to the recent endeavor of investigating residential segregation models with realistic agent behavior by studying Jump Schelling Games with agents having a single-peaked utility function. In such games, there are empty nodes in the graph and agents can strategically jump to such nodes to improve their utility. We investigate the existence of equilibria and show that they exist under specific conditions. Contrasting this, we prove that even on simple topologies like paths or rings such stable states are not guaranteed to exist. Regarding the game dynamics, we show that improving response cycles exist independently of the position of the peak in the utility function. Moreover, we show high almost tight bounds on the Price of Anarchy and the Price of Stability with respect to the recently proposed degree of integration, which counts the number of agents with a diverse neighborhood and which serves as a proxy for measuring the segregation strength. Last but not least, we show that computing a beneficial state with high integration is NP-complete and, as a novel conceptual contribution, we also show that it is NP-hard to decide if an equilibrium state can be found via improving response dynamics starting from a given initial state.
Becher, Kilian; Lagodzinski, J. A. Gregor; Parra-Arnau, Javier; Strufe, Thorsten Analysis and Prevention of Averaging Attacks Against Obfuscation ProtocolsApplied Cryptography and Network Security (ACNS), Part I 2023: 451–475
Verification and traceability of supply-chain data is a common example for public analysis of confidential data. Finding the correct balance between confidentiality and utility often is anything but trivial. In order to ensure confidentiality and thus protect companies’ competitive advantages, existing approaches employ probabilistic output obfuscation. However, it is known that this form of obfuscation might render a system subject to averaging attacks. In these attacks, an adversary repeatedly queries for the same analysis and combines the probabilistic outputs, thus implementing an estimator that eliminates the obfuscation. A clear picture on the performance of such attacks is missing, information that is crucial for mitigating averaging attacks. Our contributions are threefold: First, using an existing supply-chain verification protocol (RVP) as a particularly efficient example of protocols with output obfuscation, we extensively analyze the risk posed by averaging attacks. We prove rigorously that such attacks perform exceptionally well if obfuscation is based on random values sampled independently in every query. We generalize our analysis to all protocols that employ probabilistic output obfuscation. Second, we propose the paradigm of data-dependent deterministic obfuscation (D3O) to prevent such attacks. Third, we present mRVP, a D3O-based version of RVP, and empirically demonstrate practicality and effectiveness of D3O. The results show that our mitigations add negligible runtime overhead, do not affect accuracy, and effectively retain confidentiality.
Böther, Maximilian; Kißig, Otto; Weyand, Christopher Efficiently Computing Directed Minimum Spanning TreesSIAM Symposium on Algorithm Engineering and Experiments (ALENEX) 2023: 86–95
Computing a directed minimum spanning tree, called arborescence, is a fundamental algorithmic problem, although not as common as its undirected counterpart. In 1967, Edmonds discussed an elegant solution. It was refined to run in O(min(n2, m log n)) by Tarjan which is optimal for very dense and very sparse graphs. Gabow et al. gave a version of Edmonds’ algorithm that runs in O(n log n +m), thus asymptotically beating the Tarjan variant in the regime between sparse and dense. Despite the attention the problem received theoretically, there exists, to the best of our knowledge, no empirical evaluation of either of these algorithms. In fact, the version by Gabow et al. has never been implemented and, aside from coding competitions, all readily available Tarjan implementations run in O(n2). In this paper, we provide the first implementation of the version by Gabow et al. as well as five variants of Tarjan’s version with different underlying data structures. We evaluate these algorithms and existing solvers on a large set of real-world and random graphs.
Khomutovskiy, Ivan; Dunker, Rebekka; Dierking, Jessica; Egbert, Julian; Helms, Christian; Schöllkopf, Finn; Casel, Katrin; Fischbeck, Philipp; Friedrich, Tobias; Isaac, Davis; Krogmann, Simon; Lenzner, Pascal Applying Skeletons to Speed Up the Arc-Flags Routing AlgorithmSIAM Symposium on Algorithm Engineering and Experiments (ALENEX) 2023: 110–122
The Single-Source Shortest Path problem is classically solved by applying Dijkstra's algorithm. However, the plain version of this algorithm is far too slow for real-world applications such as routing in large road networks. To amend this, many speed-up techniques have been developed that build on the idea of computing auxiliary data in a preprocessing phase, that is used to speed up the queries. One well-known example is the Arc-Flags algorithm that is based on the idea of precomputing edge flags to make the search more goal-directed. To explain the strong practical performance of such speed-up techniques, several graph parameters have been introduced. The skeleton dimension is one such parameter that has already been used to derive runtime bounds for some speed-up techniques. Moreover, it was experimentally shown to be low in real-world road networks. We introduce a method to incorporate skeletons, the underlying structure behind the skeleton dimension, to improve routing speed-up techniques even further. As a proof of concept, we develop new algorithms called SKARF and SKARF+ that combine skeletons with Arc-Flags, and demonstrate via extensive experiments on large real-world road networks that SKARF+ yields a significant reduction of the search space and the query time of about 30% to 40% over Arc-Flags. We also prove theoretical bounds on the query time of SKARF, which is the first time an Arc-Flags variant has been analyzed in terms of skeleton dimension.
Krämer, Bastian; Stang, Moritz; Doskoč, Vanja; Schäfers, Wolfgang; Friedrich, Tobias Automated Valuation Models: Improving Model Performance by Choosing the Optimal Spatial Training LevelAmerican Real Estate Society (ARES) 2023: 1–26
The use of Automated Valuation Models (AVMs) in the context of traditional real estate valuations and their performance has been discussed in the academic community for several decades. Most studies focus on finding which method is best suited for estimating property values. One aspect that has not yet been studied scientifically is the appropriate choice of the spatial training level. The published research on AVMs usually deals with a manually defined region and fails to test the methods used on different spatial levels. The aim of our research is thus to investigate the impact of training AVM algorithms at different spatial levels in terms of valuation accuracy. We use a dataset with about 1.2 million residential properties from Germany and test four different methods, namely Ordinary Least Square, Generalized Additive Models, eXtreme Gradient Boosting and Deep Neural Network. Our results show that the right choice of spatial training level can have a major impact on the model performance, and that this impact varies across the different methods.
Jana, Satyabrata; Saha, Souvik; Sahu, Abhishek; Saurabh, Saket; Verma, Shaily Partitioning Subclasses of Chordal Graphs with Few DeletionsConference on Algorithms and Complexity (CIAC) 2023: 293–307
In the (Vertex) \(k\)-Way Cut problem, input is an undirected graph \(G\), an integer \(s\), and the goal is to find a subset \(S\) of edges (vertices) of size at most \(s\), such that \(G-S\) has at least \(k\) connected components. Downey et al. [Electr. Notes Theor. Comput. Sci. 2003] showed that \(k\)-Way Cut is W[1]-hard parameterized by \( k \). However, Kawarabayashi and Throup [FOCS 2011] showed that the problem is fixed-parameter tractable (FPT) in general graphs with respect to the parameter \(s\) and provided a \(\mathcal{O(s^s^\mathcal{O(s) n^2) \) time algorithm, where \(n \) denotes the number of vertices in \(G \). The best-known algorithm for this problem runs in time \(s^{\mathcal{O(s) n^\mathcal{O(1)}\) given by Lokshtanov et al. [ACM Tran. of Algo. 2021]. On the other hand, Vertex \(k\)-Way Cut is W[1]-hard with respect to either of the parameters, \(k\) or \(s\) or \(k+s\). These algorithmic results motivate us to look at the problems on special classes of graphs. In this paper, we consider the (Vertex) \(k\)-Way Cut problem on subclasses of chordal graphs and obtain the following results. (i) We first give a sub-exponential FPT algorithm for \(k\)-Way Cut running in time \(2^{\mathcal{O(\sqrt{s} \log s) n^\mathcal{O(1)}\) on chordal graphs. (ii) It is known that Vertex \(k\)-Way Cut is W[1]-hard on chordal graphs, in fact on split graphs, parameterized by \(k+s\). We complement this hardness result by designing polynomial-time algorithms for Vertex \(k\)-Way Cut on interval graphs, circular-arc graphs and permutation graphs.
Bläsius, Thomas; Friedrich, Tobias; Katzmann, Maximilian; Ruff, Janosch; Zeif, Ziena On the Giant Component of Geometric Inhomogeneous Random GraphsEuropean Symposium on Algorithms (ESA) 2023: 20:1–20:13
In this paper we study the threshold model of geometric inhomogeneous random graphs (GIRGs); a generative random graph model that is closely related to hyperbolic random graphs (HRGs). These models have been observed to capture complex real-world networks well with respect to the structural and algorithmic properties. Following comprehensive studies regarding their connectivity, i.e., which parts of the graphs are connected, we have a good understanding under which circumstances a giant component (containing a constant fraction of the graph) emerges. While previous results are rather technical and challenging to work with, the goal of this paper is to provide more accessible proofs. At the same time we significantly improve the previously known probabilistic guarantees, showing that GIRGs contain a giant component with probability \( 1 - exp(-Omega (n^(3-\tau )/2})) \) for graph size \(n\) and a degree distribution with power-law exponent \(tau in (2, 3)\). Based on that we additionally derive insights about the connectivity of certain induced subgraphs of GIRGs.
Casel, Katrin; Friedrich, Tobias; Schirneck, Martin; Wietheger, Simon Fair Correlation Clustering in ForestsFoundations of Responsible Computing (FORC) 2023: 9:1–9:12
The study of algorithmic fairness received growing attention recently. This stems from the awareness that bias in the input data for machine learning systems may result in discriminatory outputs. For clustering tasks, one of the most central notions of fairness is the formalization by Chierichetti, Kumar, Lattanzi, and Vassilvitskii [NeurIPS 2017]. A clustering is said to be fair, if each cluster has the same distribution of manifestations of a sensitive attribute as the whole input set. This is motivated by various applications where the objects to be clustered have sensitive attributes that should not be over- or underrepresented. Most research on this version of fair clustering has focused on centriod-based objectives. In contrast, we discuss the applicability of this fairness notion to Correlation Clustering. The existing literature on the resulting Fair Correlation Clustering problem either presents approximation algorithms with poor approximation guarantees or severely limits the possible distributions of the sensitive attribute (often only two manifestations with a 1:1 ratio are considered). Our goal is to understand if there is hope for better results in between these two extremes. To this end, we consider restricted graph classes which allow us to characterize the distributions of sensitive attributes for which this form of fairness is tractable from a complexity point of view. While existing work on Fair Correlation Clustering gives approximation algorithms, we focus on exact solutions and investigate whether there are efficiently solvable instances. The unfair version of Correlation Clustering is trivial on forests, but adding fairness creates a surprisingly rich picture of complexities. We give an overview of the distributions and types of forests where Fair Correlation Clustering turns from tractable to intractable. As the most surprising insight, we consider the fact that the cause of the hardness of Fair Correlation Clustering is not the strictness of the fairness condition. We lift most of our results to also hold for the relaxed version of the fairness condition. Instead, the source of hardness seems to be the distribution of the sensitive attribute. On the positive side, we identify some reasonable distributions that are indeed tractable. While this tractability is only shown for forests, it may open an avenue to design reasonable approximations for larger graph classes.
Jansen, Bart M. P.; Khazaliya, Liana; Kindermann, Philipp; Liotta, Giuseppe; Montecchiani, Fabrizio; Simonov, Kirill Upward and Orthogonal Planarity are W[1]-Hard Parameterized by TreewidthGraph Drawing (GD) 2023: 203–217
Upward planarity testing and Rectilinear planarity testing are central problems in graph drawing. It is known that they are both NP-complete, but XP when parameterized by treewidth. In this paper we show that these two problems are W[1]-hard parameterized by treewidth, which answers open problems posed in two earlier papers. The key step in our proof is an analysis of the All-or-Nothing Flow problem, a generalization of which was used as an intermediate step in the NP-completeness proof for both planarity testing problems. We prove that the flow problem is W[1]-hard parameterized by treewidth on planar graphs, and that the existing chain of reductions to the planarity testing problems can be adapted without blowing up the treewidth. Our reductions also show that the known \(n^{\mathcal{O(tw)}\)-time algorithms cannot be improved to run in time \(n^{o(tw)}\) unless ETH fails.
Nikfarjam, Adel; Rothenberger, Ralf; Neumann, Frank; Friedrich, Tobias Evolutionary Diversity Optimisation in Constructing Satisfying AssignmentsGenetic and Evolutionary Computation Conference (GECCO) 2023
Computing diverse solutions for a given problem, in particular evolutionary diversity optimisation (EDO), is a hot research topic in the evolutionary computation community. This paper studies the Boolean satisfiability problem (SAT) in the context of EDO. SAT is an of great importance in computer science and differs from the other problems that have been studied in EDO literature, such as KP and TSP. 1) it is possible to add more constraints (clauses) to the problem so as to forbid solutions or fix variables. 2) There can be found several powerful solvers in the literature such as minisat; we utilise such a solver to construct a diverse set of solutions. Moreover, maximising diversity provides us with invaluable information about the solution space of a given SAT problem, such as how large is the feasible region. In this study, we introduce evolutionary algorithms (EAs) employing a well-known SAT solver to maximise diversity among a set of SAT solutions explicitly. The experimental investigations indicate the introduced algorithms' capability to maximise diversity among the SAT solutions.
Li, Xiaoyue; Kötzing, Timo Experimental Analyses of Crossover on JumpGenetic and Evolutionary Computation Conference (GECCO) 2023
While it is mathematically proven that the \((\mu+1)\) GA optimizes \(\mathrm{jump}_{k}\) efficiently for low crossover probabilities, theory research still struggles with the analysis of crossover-based optimization for high crossover probabilities on this key test function. Research in this area has improved our understanding of crossover in general, in particular regarding the emergence of diversity, the crucial ingredient for successful optimization with genetic algorithms. In this paper we study the optimizing process after the \((\mu+1)\) GA has reached the plateau of \(\mathrm{jump}_{k}\). We are interested in (a) the stationary distribution of the algorithm on the plateau (when ignoring the optimum) and (b) the dynamics of the stationary distribution. We experimentally show that the \((\mu+1)\) GA achieves 10% complementary pairs if \(\mu = 10 \cdot k\), unless \(n\) is very small. Regarding the dynamics, we show samples of how bit positions gain and lose individuals with a \(0\) at that position.
Baguley, Samuel; Friedrich, Tobias; Neumann, Aneta; Neumann, Frank; Pappik, Marcus; Zeif, Ziena Fixed Parameter Multi-Objective Evolutionary Algorithms for the W-Separator ProblemGenetic and Evolutionary Computation Conference (GECCO) 2023
Parameterized analysis provides powerful mechanisms for obtaining fine-grained insights into different types of algorithms. In this work, we combine this field with evolutionary algorithms and provide parameterized complexity analysis of evolutionary multi-objective algorithms for the W-separator problem, which is a natural generalization of the vertex cover problem. The goal is to remove the minimum number of vertices such that each connected component in the resulting graph has at most W vertices. We provide different multi-objective formulations involving two or three objectives that provably lead to fixed-parameter evolutionary algorithms with respect to the value of an optimal solution OPT and W. Of particular interest are kernelizations and the reducible structures used for them. We show that in expectation the algorithms make incremental progress in finding such structures and beyond. The current best known kernelization of the W-separator uses linear programming methods and requires a non-trivial post-process to extract the reducible structures. We provide additional structural features to show that evolutionary algorithms with appropriate objectives are also capable of extracting them. Our results show that evolutionary algorithms with different objectives guide the search and admit fixed parameterized runtimes to solve or approximate (even arbitrarily close) the W-separator problem.
Friedrich, Tobias; Kötzing, Timo; Neumann, Aneta; Neumann, Frank; Radhakrishnan, Aishwarya Analysis of the (1+1) EA on LeadingOnes with ConstraintsGenetic and Evolutionary Computation Conference (GECCO ’23) 2023
Understanding how evolutionary algorithms perform on constrained problems has gained increasing attention in recent years. In this paper, we study how evolutionary algorithms optimize constrained versions of the classical LeadingOnes problem. We first provide a run time analysis for the classical (1+1)~EA on the LeadingOnes problem with a deterministic cardinality constraint, giving \($\Theta(n (n-B)\log(B) + n^2)$\) as the tight bound. Our results show that the behaviour of the algorithm is highly dependent on the constraint bound of the uniform constraint. Afterwards, we consider the problem in the context of stochastic constraints and provide insights using experimental studies on how the (\($\mu$\)+1)~EA is able to deal with these constraints in a sampling-based setting.
Bilò, Davide; Choudhary, Keerti; Cohen, Sarel; Friedrich, Tobias; Krogmann, Simon; Schirneck, Martin Fault-Tolerant ST-Diameter OraclesInternational Colloquium on Automata, Languages and Programming (ICALP) 2023: 24:1–24:20
We study the problem of estimating the \(ST\)-diameter of a graph that is subject to a bounded number of edge failures. An \(f\)-edge fault-tolerant \(ST\)-diameter oracle (\(f\)-FDO-\(ST\)) is a data structure that preprocesses a given graph \(G\), two sets of vertices \(S,T\), and positive integer \(f\). When queried with a set \(F\) of at most \(f\) edges, the oracle returns an estimate \(\widehat{D}\) of the \(ST\)-diameter \(\mathrm{diam(G-F,S,T)\), the maximum distance between vertices in \(S\) and \(T\) in \(G-F\). The oracle has stretch \(\sigma \geq 1\) if \(\mathrm{diam(G-F,S,T) leq \widehat{D leq sigma \mathrm{diam(G-F,S,T)\). If \(S\) and \(T\) both contain all vertices, the data structure is called an \(f\)-edge fault-tolerant diameter oracle (\(f\)-FDO). An \(f\)-edge fault-tolerant distance sensitivity oracles (\(f\)-DSO) estimates the pairwise graph distances under up to \(f\) failures. We design new \(f\)-FDOs and \(f\)-FDO-\(ST\)s by reducing their construction to that of all-pairs and single-source \(f\)-DSOs. We obtain several new tradeoffs between the size of the data structure, stretch guarantee, query and preprocessing times for diameter oracles by combining our black-box reductions with known results from the literature. We also provide an information-theoretic lower bound on the space requirement of approximate \(f\)-FDOs. We show that there exists a family of graphs for which any \(f\)-FDO with sensitivity \(f \ge 2\) and stretch less than \(5/3\) requires \(\Omega(n^{3/2})\) bits of space, regardless of the query time.
Fomin, Fedor; Golovach, Petr; Sagunov, Danil; Simonov, Kirill Approximating Long Cycle Above Dirac’s GuaranteeInternational Colloquium on Automata, Languages and Programming (ICALP) 2023: 60:1–60:18
Parameterization above (or below) a guarantee is a successful concept in parameterized algorithms. The idea is that many computational problems admit “natural” guarantees bringing to algorithmic questions whether a better solution (above the guarantee) could be obtained efficiently. For example, for every boolean CNF formula on m clauses, there is an assignment that satisfies at least m/2 clauses. How difficult is it to decide whether there is an assignment satisfying more than m/2 + k clauses? Or, if an n-vertex graph has a perfect matching, then its vertex cover is at least n/2. Is there a vertex cover of size at least n/2 + k for some \(k \geq 1\) and how difficult is it to find such a vertex cover? The above guarantee paradigm has led to several exciting discoveries in the areas of parameterized algorithms and kernelization. We argue that this paradigm could bring forth fresh perspectives on well-studied problems in approximation algorithms. Our example is the longest cycle problem. One of the oldest results in extremal combinatorics is the celebrated Dirac’s theorem from 1952. Dirac’s theorem provides the following guarantee on the length of the longest cycle: for every 2-connected n-vertex graph G with minimum degree \(\delta(G) \leq n/2\), the length of a longest cycle L is at least \(2\delta(G)\). Thus the “essential” part in finding the longest cycle is in approximating the “offset” \(k = L − 2\delta(G)\). The main result of this paper is the above-guarantee approximation theorem for k. Informally, the theorem says that approximating the offset k is not harder than approximating the total length L of a cycle. In other words, for any (reasonably well-behaved) function f, a polynomial time algorithm constructing a cycle of length f(L) in an undirected graph with a cycle of length L, yields a polynomial time algorithm constructing a cycle of length \(2\delta(G) + \Omega(f(k))\).
Friedrich, Tobias; Göbel, Andreas; Katzmann, Maximilian; Schiller, Leon Cliques in High-Dimensional Geometric Inhomogeneous Random GraphsInternational Colloquium on Automata, Languages and Programming (ICALP) 2023: 62:1–62:13
A recent trend in the context of graph theory is to bring theoretical analyses closer to empirical observations, by focusing the studies on random graph models that are used to represent practical instances. There, it was observed that geometric inhomogeneous random graphs (GIRGs) yield good representations of complex real-world networks, by expressing edge probabilities as a function that depends on (heterogeneous) vertex weights and distances in some underlying geometric space that the vertices are distributed in. While most of the parameters of the model are understood well, it was unclear how the dimensionality of the ground space affects the structure of the graphs. In this paper, we complement existing research into the dimension of geometric random graph models and the ongoing study of determining the dimensionality of real-world networks, by studying how the structure of GIRGs changes as the number of dimensions increases. We prove that, in the limit, GIRGs approach non-geometric inhomogeneous random graphs and present insights on how quickly the decay of the geometry impacts important graph structures. In particular, we study the expected number of cliques of a given size as well as the clique number and characterize phase transitions at which their behavior changes fundamentally. Finally, our insights help in better understanding previous results about the impact of the dimensionality on geometric random graphs.
Bilò, Davide; Cohen, Sarel; Friedrich, Tobias; Gawendowicz, Hans; Klodt, Nicolas; Lenzner, Pascal; Skretas, George Temporal Network Creation GamesInternational Joint Conference on Artificial Intelligence (IJCAI) 2023: 2511–2519
Most networks are not static objects, but instead they change over time. This observation has sparked rigorous research on temporal graphs within the last years. In temporal graphs, we have a fixed set of nodes and the connections between them are only available at certain time steps. This gives rise to a plethora of algorithmic problems on such graphs, most prominently the problem of finding temporal spanners, i.e., the computation of subgraphs that guarantee all pairs reachability via temporal paths. To the best of our knowledge, only centralized approaches for the solution of this problem are known. However, many real-world networks are not shaped by a central designer but instead they emerge and evolve by the interaction of many strategic agents. This observation is the driving force of the recent intensive research on game-theoretic network formation models. In this work we bring together these two recent research directions: temporal graphs and game-theoretic network formation. As a first step into this new realm, we focus on a simplified setting where a complete temporal host graph is given and the agents, corresponding to its nodes, selfishly create incident edges to ensure that they can reach all other nodes via temporal paths in the created network. This yields temporal spanners as equilibria of our game. We prove results on the convergence to and the existence of equilibrium networks, on the complexity of finding best agent strategies, and on the quality of the equilibria. By taking these first important steps, we uncover challenging open problems that call for an in-depth exploration of the creation of temporal graphs by strategic agents.
Bilò, Davide; Bilò, Vittorio; Döring, Michelle; Lenzner, Pascal; Molitor, Louise; Schmidt, Jonas Schelling Games with Continuous TypesInternational Joint Conference on Artificial Intelligence (IJCAI) 2023: 2520–2527
In most major cities and urban areas, residents form homogeneous neighborhoods along ethnic or socioeconomic lines. This phenomenon is widely known as residential segregation and has been studied extensively. Fifty years ago, Schelling proposed a landmark model that explains residential segregation in an elegant agent-based way. A recent stream of papers analyzed Schelling’s model using game-theoretic approaches. However, all these works considered models with a given number of discrete types modeling different ethnic groups. We focus on segregation caused by non-categorical attributes, such as household income or position in a political left-right spectrum. For this, we consider agent types that can be represented as real numbers. This opens up a great variety of reasonable models and, as a proof of concept, we focus on several natural candidates. In particular, we consider agents that evaluate their location by the average type-difference or the maximum type-difference to their neighbors, or by having a certain tolerance range for type-values of neighboring agents. We study the existence and computation of equilibria and provide bounds on the Price of Anarchy and Stability. Also, we present simulation results that compare our models and shed light on the obtained equilibria for our variants.
Gadea Harder, Jonathan; Krogmann, Simon; Lenzner, Pascal; Skopalik, Alexander Strategic Resource Selection with Homophilic AgentsInternational Joint Conference on Artificial Intelligence (IJCAI) 2023: 2701–2709
The strategic selection of resources by selfish agents is a classic research direction, with Resource Selection Games and Congestion Games as prominent examples. In these games, agents select available resources and their utility then depends on the number of agents using the same resources. This implies that there is no distinction between the agents, i.e., they are anonymous. We depart from this very general setting by proposing Resource Selection Games with heterogeneous agents that strive for joint resource usage with similar agents. So, instead of the number of other users of a given resource, our model considers agents with different types and the decisive feature is the fraction of same-type agents among the users. More precisely, similarly to Schelling Games, there is a tolerance threshold \(\tau \in [0,1]\) which specifies the agents' desired minimum fraction of same-type agents on a resource. Agents strive to select resources where at least a \(\tau\)-fraction of those resources' users have the same type as themselves. For \(\tau=1\), our model generalizes Hedonic Diversity Games with a peak at \(1\). For our general model, we consider the existence and quality of equilibria and the complexity of maximizing social welfare. Additionally, we consider a bounded rationality model, where agents can only estimate the utility of a resource, since they only know the fraction of same-type agents on a given resource, but not the exact numbers. Thus, they cannot know the impact a strategy change would have on a target resource. Interestingly, we show that this type of bounded rationality yields favorable game-theoretic properties and specific equilibria closely approximate equilibria of the full knowledge setting.
Deligkas, Argyrios; Eiben, Eduard; Skretas, George Minimizing Reachability Times on Temporal Graphs via Shifting LabelsInternational Joint Conference on Artificial Intelligence (IJCAI) 2023: 5333–5340
We study how we can accelerate the spreading of information in temporal graphs via delaying operations; a problem that captures real-world applications varying from information flows to distribution schedules. In a temporal graph there is a set of fixed vertices and the available connections between them change over time in a predefined manner. We observe that, in some cases, the delay of some connections can in fact decrease the time required to reach from some vertex (source) to another vertex (target). We study how we can minimize the maximum time a set of source vertices needs to reach every other vertex of the graph when we are allowed to delay some of the connections of the graph. For one source, we prove that the problem is W[2]-hard and NP-hard, when parameterized by the number of allowed delays. On the other hand, we derive a polynomial-time algorithm for one source and unbounded number of delays. This is the best we can hope for; we show that the problem becomes NP-hard when there are two sources and the number of delays is not bounded. We complement our negative result by providing an FPT algorithm parameterized by the treewidth of the graph plus the lifetime of the optimal solution. Finally, we provide polynomial-time algorithms for several classes of graphs.
Ganian, Robert; Khazaliya, Liana; Simonov, Kirill Consistency Checking Problems: A Gateway to Parameterized Sample ComplexityInternational Symposium on Parameterized and Exact Computation (IPEC) 2023: 18:1–18:17
Recently, Brand, Ganian and Simonov introduced a parameterized refinement of the classical PAC-learning sample complexity framework. A crucial outcome of their investigation is that for a very wide range of learning problems, there is a direct and provable correspondence between fixed-parameter PAC-learnability (in the sample complexity setting) and the fixed-parameter tractability of a corresponding “consistency checking” search problem (in the setting of computational complexity). The latter can be seen as generalizations of classical search problems where instead of receiving a single instance, one receives multiple yes- and no-examples and is tasked with finding a solution which is consistent with the provided examples. Apart from a few initial results, consistency checking problems are almost entirely unexplored from a parameterized complexity perspective. In this article, we provide an overview of these problems and their connection to parameterized sample complexity, with the primary aim of facilitating further research in this direction. Afterwards, we establish the fixed-parameter (in)-tractability for some of the arguably most natural consistency checking problems on graphs, and show that their complexity-theoretic behavior is surprisingly very different from that of classical decision problems. Our new results cover consistency checking variants of problems as diverse as (k-)Path, Matching, 2-Coloring, Independent Set and Dominating Set, among others.
Khazaliya, Liana; Kindermann, Philipp; Liotta, Giuseppe; Montecchiani, Fabrizio; Simonov, Kirill The st-Planar Edge Completion Problem Is Fixed-Parameter TractableInternational Symposium Algorithms and Computation (ISAAC) 2023: 46:1–46:13
The problem of deciding whether a biconnected planar digraph \(G = (V, E)\) can be augmented to become an st-planar graph by adding a set of oriented edges \(E′ \subseteq V \times V\) is known to be NP-complete. We show that the problem is fixed-parameter tractable when parameterized by the size of the set \(E′\).
Ashok, Pradeesha; Das, Sayani; Kanesh, Lawqueen; Saurabh, Saket; Tomar, Avi; Verma, Shaily Burn and WinInternational Workshop on Combinatorial Algorithms (IWOCA) 2023: 36–48
Given a graph \(G\) and an integer \(k\), the graph burning problem asks whether the graph \(G\) can be burned in at most \(k\) rounds. Graph burning is a model for information spreading in a network, where we study how fast the information spreads in the network through its vertices. In each round, the fire is started at an unburned vertex and the fire spreads to all its neighbors in the next round, burning all of them and so on. The minimum number of rounds required to burn the whole graph \(G\) is called the burning number of \(G\). Graph burning problem is NP-hard even for the union of disjoint paths. Moreover, the problem is known to be W[1]-hard when parameterized by the burning number and para-NP-hard when parameterized by treewidth. In this paper, we give a fixed-parameter tractable (FPT) algorithm for graph burning problem when parameterized by treewidth and burning number of the graph. Y. Kobayashi and Y. Otachi [Algorithmica 2022] proved that the problem is FPT parameterized by distance to cographs and gave a double exponential time FPT algorithm when parameterized by distance to split graphs. We improve these results partially and give an FPT algorithm for the problem when parameterized by distance to cographs \(\cap\) split graphs (threshold graphs) that runs in \(2^{\mathcal{O(k\ln k)}\). We also design a kernel for the problem in trees. Furthermore, we give an exact algorithm to find the burning number of a graph that runs in time \(4^n n^\mathcal{O(1)}\), where \(n\) is the number of vertices in the input graph.
Bhyravarapu, Sriram; Jana, Satyabrata; Kanesh, Lawqueen; Saurabh, Saket; Verma, Shaily Parameterized Algorithms for Eccentricity Shortest Path ProblemInternational Workshop on Combinatorial Algorithms (IWOCA) 2023: 74–86
Given an undirected graph \(G=(V,E) \) and an integer \(\ell\), the Eccentricity Shortest Path Problem (ESP) problem asks to check if there exists a shortest path \(P\) such that for every vertex \(vin V(G)\), there is a vertex \(w\in P\) such that \(d_G(v,w)\leq \ell\), where \(d_G(v,w)\) represents the distance between \(v\) and \(w\) in \(G\). Dragan and Leitert [Theor. Comput. Sci. 2017] studied the optimization version of this problem which asks to find the minimum \(\ell\) for ESP and showed that it is sf NP-hard even on planar bipartite graphs with maximum degree 3. They also showed that ESP is sf W[2]-hard when parameterized by \(\ell \). On the positive side, Kucera and Suchy [IWOCA 2021] showed that ESP is fixed-parameter tractable (FPT) when parameterized by modular width, cluster vertex deletion set, maximum leaf number, or the combined parameters disjoint paths deletion set and \( \ell \). It was asked as an open question in the same paper, if ESP is FPT parameterized by disjoint paths deletion set or feedback vertex set. We answer these questions and obtain the following results. We prove that ESP is FPT when parameterized by disjoint paths deletion set, split vertex deletion set, or the combined parameters feedback vertex set and \(\ell\). Moreover, we design a \((1+\epsilon)\)-factor FPT approximation algorithm when parameterized by the feedback vertex set number. We also show that ESP is W[2]-hard when parameterized by the chordal vertex deletion set.
Cohen, Sarel; Fischbeck, Philipp; Friedrich, Tobias; Krejca, Martin S. The Common-Neighbors Metric is Noise-Robust and Reveals Substructures of Real-World NetworksPacific-Asia Conference on Knowledge Discovery and Data Mining (PAKDD) 2023: 67–79
Real-world networks typically display a complex structure that is hard to explain by a single model. A common approach is to partition the edges of the network into disjoint simpler structures. An important property in this context is locality—incident vertices usually have many common neighbors. This allows to classify edges into two groups, based on the number of the common neighbors of their incident vertices. Formally, this is captured by the common-neighbors (CN) metric, which forms the basis of many metrics for detecting outlier edges. Such outliers can be interpreted as noise or as a substructure. We aim to understand how useful the metric is, and empirically analyze several scenarios. We randomly insert outlier edges into real-world and generated graphs with high locality, and measure the metric accuracy for partitioning the combined edges. In addition, we use the metric to decompose real-world networks, and measure properties of the partitions. Our results show that the CN metric is a very good classifier that can reliably detect noise up to extreme levels (\(83\%\) noisy edges). We also provide mathematically rigorous analyses on special random-graph models. Last, we find the CN metric consistently decomposes real-world networks into two graphs with very different structures.
Friedrich, Tobias; Gawendowicz, Hans; Lenzner, Pascal; Zahn, Arthur The Impact of Cooperation in Bilateral Network CreationACM Symposium on Principles of Distributed Computing (PODC) 2023
Many real-world networks, like the Internet or social networks, are not the result of central design but instead the outcome of the interaction of local agents that selfishly optimize their individual utility. The well-known Network Creation Game by Fabrikant, Luthra, Maneva, Papadimitriou, and Shenker [PODC 2003] models this. There, agents corresponding to network nodes buy incident edges towards other agents for a price of \(\alpha > 0\) and simultaneously try to minimize their buying cost and their total hop distance. Since in many real-world networks, e.g., social networks, consent from both sides is required to establish and maintain a connection, Corbo and Parkes [PODC 2005] proposed a bilateral version of the Network Creation Game, in which mutual consent and payment are required in order to create edges. It is known that this cooperative version has a significantly higher Price of Anarchy compared to the unilateral version. On the first glance this is counter-intuitive, since cooperation should help to avoid socially bad states. However, in the bilateral version only a very restrictive form of cooperation is considered. We investigate this trade-off between the amount of cooperation and the Price of Anarchy by analyzing the bilateral version with respect to various degrees of cooperation among the agents. With this, we provide insights into what kind of cooperation is needed to ensure that socially good networks are created. As a first step in this direction, we focus on tree networks and present a collection of asymptotically tight bounds on the Price of Anarchy that precisely map the impact of cooperation. Most strikingly, we find that weak forms of cooperation already yield a significantly improved Price of Anarchy. In particular, the cooperation of coalitions of size 3 is enough to achieve constant bounds. Moreover, for general networks we show that enhanced cooperation yields close to optimal networks for a wide range of edge prices. Along the way, we disprove an old conjecture by Corbo and Parkes [PODC 2005].
Anand, Konrad; Göbel, Andreas; Pappik, Marcus; Perkins, Will Perfect Sampling for Hard Spheres from Strong Spatial MixingInternational Conference on Randomization and Computation (Random) 2023: 38:1–38:18
We provide a perfect sampling algorithm for the hard-sphere model on subsets of ℝd with expected running time linear in the volume under the assumption of strong spatial mixing. A large number of perfect and approximate sampling algorithms have been devised to sample from the hard-sphere model, and our perfect sampling algorithm is efficient for a range of parameters for which only efficient approximate samplers were previously known and is faster than these known approximate approaches. Our methods also extend to the more general setting of Gibbs point processes interacting via finite-range, repulsive potentials.
Angrick, Sebastian; Bals, Ben; Casel, Katrin; Cohen, Sarel; Friedrich, Tobias; Hastrich, Niko; Hradilak, Theresa; Issac, Davis; Kißig, Otto; Schmidt, Jonas; Wendt, Leo Solving Directed Feedback Vertex Set by Iterative Reduction to Vertex CoverSymposium on Experimental Algorithms (SEA) 2023: 10:1–10:14
In the Directed Feedback Vertex Set (DFVS) problem, one is given a directed graph \(G = (V,E)\) and wants to find a minimum cardinality set \(S \subseteq V\) such that \(G-S\) is acyclic. DFVS is a fundamental problem in computer science and finds applications in areas such as deadlock detection. The problem was the subject of the 2022 PACE coding challenge. We develop a novel exact algorithm for the problem that is tailored to perform well on instances that are mostly bi-directed. For such instances, we adapt techniques from the well-researched vertex cover problem. Our core idea is an iterative reduction to vertex cover. To this end, we also develop a new reduction rule that reduces the number of not bi-directed edges. With the resulting algorithm, we were able to win third place in the exact track of the PACE challenge. We perform computational experiments and compare the running time to other exact algorithms, in particular to the winning algorithm in PACE. Our experiments show that we outpace the other algorithms on instances that have a low density of uni-directed edges.
Fomin, Fedor V.; Golovach, Petr A.; Korhonen, Tuukka; Simonov, Kirill; Stamoulis Giannοs Fixed-Parameter Tractability of Maximum Colored Path and BeyondSymposium on Discrete Algorithms (SODA) 2023
We introduce a general method for obtaining fixed-parameter algorithms for problems about finding paths in undirected graphs, where the length of the path could be unbounded in the parameter. The first application of our method is a randomized algorithm, that given a colored \($n$\)-vertex undirected graph, vertices \($s$\) and \($t$\), and an integer \($k$\), finds an \($(s,t)$\)-path containing at least \($k$\) different colors in time \($2^k n^\mathcal{O(1)}$\). This is the first FPT algorithm for this problem, and it generalizes the algorithm of Björklund, Husfeldt, and Taslaman~[SODA~2012] on finding a path through \($k$\) specified vertices. It also implies the first \($2^k n^\mathcal{O(1)}$\) time algorithm for finding an \($(s,t)$\)-path of length at least \($k$\). Our method yields FPT algorithms for even more general problems. For example, we consider the problem where the input consists of an \($n$\)-vertex undirected graph \($G$\), a matroid \($M$\) whose elements correspond to the vertices of \($G$\) and which is represented over a finite field of order \($q$\), a positive integer weight function on the vertices of \($G$\), two sets of vertices \($S,T subseteq V(G)$\), and integers \($p,k,w$\), and the task is to find \($p$\) vertex-disjoint paths from \($S$\) to \($T$\) so that the union of the vertices of these paths contains an independent set of \($M$\) of cardinality \($k$\) and weight \($w$\), while minimizing the sum of the lengths of the paths. We give a \($2^p+\mathcal{O(k^2 log (q+k)) n^\mathcal{O w$\) time randomized algorithm for this problem.
Bläsius, Thomas; Friedrich, Tobias; Katzmann, Maximilian; Stephan, Daniel Strongly Hyperbolic Unit Disk GraphsSymposium Theoretical Aspects of Computer Science (STACS) 2023: 13:1–13:17
The class of Euclidean unit disk graphs is one of the most fundamental and well-studied graph classes with underlying geometry. In this paper, we identify this class as a special case in the broader class of hyperbolic unit disk graphs and introduce strongly hyperbolic unit disk graphs as a natural counterpart to the Euclidean variant. In contrast to the grid-like structures exhibited by Euclidean unit disk graphs, strongly hyperbolic networks feature hierarchical structures, which are also observed in complex real-world networks. We investigate basic properties of strongly hyperbolic unit disk graphs, including adjacencies and the formation of cliques, and utilize the derived insights to demonstrate that the class is useful for the development and analysis of graph algorithms. Specifically, we develop a simple greedy routing scheme and analyze its performance on strongly hyperbolic unit disk graphs in order to prove that routing can be performed more efficiently on such networks than in general.
Friedrich, Tobias; Issac, Davis; Kumar, Nikhil; Mallek, Nadym; Zeif, Ziena Approximate Max-Flow Min-Multicut Theorem for Graphs of Bounded TreewidthSymposium Theory of Computing (STOC) 2023: 1325–1334
We prove an approximate max-multiflow min-multicut theorem for bounded treewidth graphs. In particular, we show the following: Given a treewidth-\(r\) graph, there exists a (fractional) multicommodity flow of value \(f\), and a multicut of capacity \(c\) such that \(f leq c leq \mathcal{O(\ln(r + 1)) \cdot f \) . It is well known that the multiflow-multicut gap on an \(r\)-vertex (constant degree) expander graph can be \(\Omega(ln r)\), and hence our result is tight up to constant factors. Our proof is constructive, and we also obtain a polynomial time \( \mathcal{O(ln(r + 1))\)-approximation algorithm for the minimum multicut problem on treewidth-r graphs. Our algorithm proceeds by rounding the optimal fractional solution to the natural linear programming relaxation of the multicut problem. We introduce novel modifications to the well-known region growing algorithm to facilitate the rounding while guaranteeing at most a logarithmic factor loss in the treewidth.
Bilò, Davide; Chechik, Shiri; Choudhary, Keerti; Cohen, Sarel; Friedrich, Tobias; Krogmann, Simon; Schirneck, Martin Approximate Distance Sensitivity Oracles in Subquadratic SpaceSymposium on Theory of Computing (STOC) 2023: 1396–1409
An \(f\)-edge fault-tolerant distance sensitive oracle (\(f\)-DSO) with stretch \(\sigma\ge1\) is a data structure that preprocesses a given undirected, unweighted graph \(G\) with \(n\) vertices and \(m\) edges, and a positive integer \(f\). When queried with a pair of vertices \(s,t\) and a set \(F\) of at most \(f\) edges, it returns a \(\sigma\)-approximation of the \(s\)-\(t\)-distance in \(G-F\). We study \(f\)-DSOs that take subquadratic space. Thorup and Zwick [JACM 2015] showed that this is only possible for \(\sigma\geq3\). We present, for any constant \(f\geq1\) and \(\alpha\in(0,\frac{1}{2})\), and any \(\varepsilon>0\), an \(f\)-DSO with stretch \(3+\varepsilon\) that takes \(\tilde{O}(n^{2-\frac{\alpha}{f+1}}/\varepsilon)\cdot{}O(\log n/\varepsilon)^{f+1}\) space and has an \(O(n^\alpha/\varepsilon^2)\) query time. We also give an improved construction for graphs with diameter at most \(D\). For any constant \(k\), we devise an \(f\)-DSO with stretch \(2k-1\) that takes \(O(D^{f+o(1)}n^{1+1/k})\) space and has \(\tilde{O}(D^o(1)})\) query time, with a preprocessing time of \(O(D^{f+o(1)}mn^{1/k})\). Chechik, Cohen, Fiat, and Kaplan [SODA 2017] presented an \(f\)-DSO with stretch \(1{+}\varepsilon\) and preprocessing time \(\tilde{O}_\varepsilon(n^5)\), albeit with a super-quadratic space requirement. We show how to reduce their preprocessing time to \(O(mn^{2})\cdot{}O(\log n/\varepsilon)^{f}\).
Baligács, Júlia; Disser, Yann; Soheil, Farehe; Weckbecker, David Tight Analysis of the Lazy Algorithm for Open Online Dial-a-RideWorkshop Algorithms and Data Structures (WADS) 2023: 43–64
In the open online dial-a-ride problem, a single server has to deliver transportation requests appearing over time in some metric space, subject to minimizing the completion time. We improve on the best known upper bounds on the competitive ratio on general metric spaces and on the half-line, for both the preemptive and non-preemptive version of the problem. We achieve this by revisiting the algorithm Lazy recently suggested in [WAOA, 2022] and giving an improved and tight analysis. More precisely, we show that it has competitive ratio 2.457 on general metric spaces and 2.366 on the half-line. This is the first upper bound that beats known lower bounds of 2.5 for schedule-based algorithms as well as the natural Replan algorithm.
Bandyapadhyay, Sayan; Fomin, Fedor V.; Inamdar, Tanmay; Panolan, Fahad; Simonov, Kirill Socially Fair Matching: Exact and Approximation AlgorithmsWorkshop on Algorithms and Data Structures (WADS) 2023: 79–92
Matching problems are some of the most well-studied problems in graph theory and combinatorial optimization, with a variety of theoretical as well as practical motivations. However, in many applications of optimization problems, a ``solution'' corresponds to real-life decisions that have major impact on humans belonging to diverse groups defined by attributes such as gender, race, or ethnicity. Due to this motivation, the notion of \emph{algorithmic fairness has recently emerged to prominence. Depending on specific application, researchers have introduced several notions of fairness. In this paper, we study a problem called ``Socially Fair Matching'', which combines the traditional Minimum Weight Perfect Matching problem with the notion of social fairness that has been studied in clustering literature [Abbasi et al., and Ghadiri et al., FAccT, 2021]. In our problem, the input is an edge-weighted complete bipartite graph, where the bipartition represent two groups of entities. The goal is to find a perfect matching as well as an assignment that assigns the cost of each matched edge to one of its endpoints, such that the maximum of the total cost assigned to either of the two groups is minimized. Unlike Minimum Weight Perfect Matching, we show that Socially Fair Matching is weakly NP-hard. On the positive side, we design a deterministic PTAS for the problem when the edge weights are arbitrary. Furthermore, if the weights are integers and polynomial in the number of vertices, then we give a randomized polynomial-time algorithm that solves the problem exactly. Next, we show that this algorithm can be used to obtain a randomized FPTAS when the weights are arbitrary.
Bilò, Davide; Choudhary, Keerti; Cohen, Sarel; Friedrich, Tobias; Krogmann, Simon; Schirneck, Martin Compact Distance Oracles with Large Sensitivity and Low StretchAlgorithms and Data Structures Symposium (WADS) 2023: 149–163
An \(f\)-edge fault-tolerant distance sensitive oracle (\(f\)-DSO) with stretch \(\sigma \geq 1\) is a data-structure that preprocesses an input graph \(G = (V,E)\). When queried with the triple \((s,t,F)\), where \(s, t \in V\) and \(F \subseteq E\) contains at most \(f\) edges of \(G\), the oracle returns an estimate \(\widehat{d_G-F(s,t)\) of the distance \(d_{G-F}(s,t)\) between \(s\) and \(t\) in the graph \(G-F\) such that \(d_{G-F}(s,t) leq \widehat{d_G-F(s,t) leq sigma cdot d_G-F(s,t)\). For any positive integer \(k \ge 2\) and any \(0 < \alpha < 1\), we present an \(f\)-DSO with sensitivity \(f = o(\log n/\log\log n)\), stretch \(2k-1\), space \(O(n^{1+\frac{1}{k+\alpha+o(1)})\), and an \(\widetildeO(n^1+\frac{1}{k - \frac{\alpha}{k(f+1)}})\) query time. Prior to our work, there were only three known \(f\)-DSOs with subquadratic space. The first one by Chechik et al. [Algorithmica 2012] has a stretch of \((8k-2)(f+1)\), depending on \(f\). Another approach is storing an \(f\)-edge fault-tolerant \((2k-1)\)-spanner of \(G\). The bottleneck is the large query time due to the size of any such spanner, which is \(\Omega(n^{1+1/k})\) under the Erdős girth conjecture. Bilò et al. [STOC 2023] gave a solution with stretch \(3+\varepsilon\), query time \(O(n^{\alpha})\) but space \(O(n^{2-\frac{\alpha}{f+1}})\), approaching the quadratic barrier for large sensitivity. In the realm of subquadratic space, our \(f\)-DSOs are the first ones that guarantee, at the same time, large sensitivity, low stretch, and non-trivial query time. To obtain our results, we use the approximate distance oracles of Thorup and Zwick [JACM 2005], and the derandomization of the \(f\)-DSO of Weimann and Yuster [TALG 2013] that was recently given by Karthik and Parter [SODA 2021].
Bandyapadhyay, Sayan; Fomin, Fedor V.; Inamdar, Tanmay; Simonov, Kirill Proportionally Fair Matching with Multiple GroupsWorkshop on Graph-Theoretic Concepts in Computer Science (WG) 2023: 1–15
We study matching problems with the notion of proportional fairness. Proportional fairness is one of the most popular notions of group fairness where every group is represented up to an extent proportional to the final selection size. Matching with proportional fairness or more commonly, proportionally fair matching, was introduced in [Chierichetti et al., AISTATS, 2019]. In this problem, we are given a graph \($G$\) whose edges are colored with colors from a set \($C$\). The task is for given \($0\le \alpha \le \beta \le 1$\), to find a maximum \($(\alpha,\beta)$\)-balanced matching \($M$\) in \($G$\), that is a matching where for every color \($c\in C$\) the number of edges in \($M$\) of color \($c$\) is between \($\alpha|M|$\) and \($\beta |M|$\). Chierichetti et al. initiated the study of this problem with two colors and in the context of bipartite graphs only. However, in many practical applications, the number of colors---although often a small constant---is larger than two. In this work, we make the first step towards understanding the computational complexity of proportionally fair matching with more than two colors. We design exact and approximation algorithms achieving reasonable guarantees on the quality of the matching as well as on the time complexity, and our algorithms work in general graphs. Our algorithms are also supported by suitable hardness bounds.
Casel, Katrin; Friedrich, Tobias; Issac, Davis; Niklanovits, Aikaterini; Zeif, Ziena Efficient Constructions for the Gyori-Lovasz Theorem on Almost Chordal GraphsWorkshop Graph-Theoretic Concepts in Computer Science (WG) 2023: 143–156
In the 1970s, GyHo}ri and Lovász showed that for a \(k\)-connected \(n\)-vertex graph, a given set of terminal vertices \(t_1, \dots, t_k\) and natural numbers \(n_1, \dots, n_k\) satisfying \(\sum_{i=1^k n_i = n\), a connected vertex partition \(S_1, \dots, S_k\) satisfying \(t_i \in S_i\) and \(|S_i| = n_i\) exists. However, polynomial algorithms to actually compute such partitions are known so far only for \(k \leq 4\). This motivates us to take a new approach and constrain this problem to particular graph classes instead of restricting the values of \(k\). More precisely, we consider \(k\)-connected chordal graphs and a broader class of graphs related to them. For the first class, we give an algorithm with \(\mathcal{O(n^2)\) running time that solves the problem exactly, and for the second, an algorithm with \(\mathcal{O(n^4)\) running time that deviates on at most one vertex from the required vertex partition sizes.
Fomin, Fedor; Golovach, Petr; Sagunov, Danil; Simonov, Kirill Turán’s Theorem Through Algorithmic LensWorkshop on Graph-Theoretic Concepts in Computer Science (WG) 2023: 348–362
The fundamental theorem of Turán from Extremal Graph Theory determines the exact bound on the number of edges \($t_r(n)$\) in an \($n$\)-vertex graph that does not contain a clique of size \($r+1$\). We establish an interesting link between Extremal Graph Theory and Algorithms by providing a simple compression algorithm that in linear time reduces the problem of finding a clique of size \($\ell$\) in an \($n$\)-vertex graph \($G$\) with \($m ge t_r(n)-k$\) edges, where \($\ell\leq r+1$\), to the problem of finding a maximum clique in a graph on at most \($5k$\) vertices. This also gives us an algorithm deciding in time \($2.49^k\cdot(n + m)$\) whether \($G$\) has a clique of size \($\ell$\). As a byproduct of the new compression algorithm, we give an algorithm that in time \($2^\mathcal{O(td^2 ) \cdot n^2$\) decides whether a graph contains an independent set of size at least \($n/(d+1) +t$\). Here \($d$\) is the average vertex degree of the graph \($G$\). The multivariate complexity analysis based on ETH indicates that the asymptotical dependence on several parameters in the running times of our algorithms is tight.
