2024 [ nach oben ]

Constantinescu, Andrei; Lenzner, Pascal; Reiffenhäuser, Rebecca; Schmand, Daniel; Varricchio, Giovanna Solving Woeginger’s Hiking Problem: Wonderful Partitions in Anonymous Hedonic GamesInternational Colloquium on Automata, Languages and Programming (ICALP) 2024
[ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
 
A decade ago, Gerhard Woeginger posed an open problem that became well-known as "Woeginger’s Hiking Problem": Consider a group of n people that want to go hiking; everyone expresses preferences over the size of their hiking group in the form of an interval between 1 and n. Is it possible to efficiently assign the n people to a set of hiking subgroups so that every person approves the size of their assigned subgroup? The problem is also known as efficiently deciding if an instance of an anonymous Hedonic Game with interval approval preferences admits a wonderful partition. We resolve the open problem in the affirmative by presenting an O(n⁵) time algorithm for Woeginger’s Hiking Problem. Our solution is based on employing a dynamic programming approach for a specific rectangle stabbing problem from computational geometry. Moreover, we propose natural, more demanding extensions of the problem, e.g., maximizing the number of satisfied participants and variants with single-peaked preferences, and show that they are also efficiently solvable. Last but not least, we employ our solution to efficiently compute a partition that maximizes the egalitarian welfare for anonymous single-peaked Hedonic Games.
@inproceedings{constantinescu2024solving, abstract = {A decade ago, Gerhard Woeginger posed an open problem that became well-known as "Woeginger’s Hiking Problem": Consider a group of n people that want to go hiking; everyone expresses preferences over the size of their hiking group in the form of an interval between 1 and n. Is it possible to efficiently assign the n people to a set of hiking subgroups so that every person approves the size of their assigned subgroup? The problem is also known as efficiently deciding if an instance of an anonymous Hedonic Game with interval approval preferences admits a wonderful partition. We resolve the open problem in the affirmative by presenting an O(n⁵) time algorithm for Woeginger’s Hiking Problem. Our solution is based on employing a dynamic programming approach for a specific rectangle stabbing problem from computational geometry. Moreover, we propose natural, more demanding extensions of the problem, e.g., maximizing the number of satisfied participants and variants with single-peaked preferences, and show that they are also efficiently solvable. Last but not least, we employ our solution to efficiently compute a partition that maximizes the egalitarian welfare for anonymous single-peaked Hedonic Games.}, author = {Constantinescu, Andrei and Lenzner, Pascal and Reiffenhäuser, Rebecca and Schmand, Daniel and Varricchio, Giovanna}, booktitle = {International Colloquium on Automata, Languages and Programming (ICALP)}, keywords = {icalp pascallenzner year2024}, title = {Solving Woeginger's Hiking Problem: Wonderful Partitions in Anonymous Hedonic Games}, year = 2024 }

2023 [ nach oben ]

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
[ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
 
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 \(2\delta(G) + \Omega(f(k))\).
@inproceedings{fomin2023approximating, abstract = {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 \(2\delta(G) + \Omega(f(k))\).}, author = {Fomin, Fedor and Golovach, Petr and Sagunov, Danil and Simonov, Kirill}, booktitle = {International Colloquium on Automata, Languages and Programming (ICALP)}, keywords = {sys:relevantfor:ae icalp kirillsimonov year2023}, pages = {60:1–60:18}, title = {Approximating Long Cycle Above Dirac’s Guarantee}, year = 2023 }

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
[ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
 
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.
@inproceedings{friedrich2023cliques, abstract = {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.}, author = {Friedrich, Tobias and Göbel, Andreas and Katzmann, Maximilian and Schiller, Leon}, booktitle = {International Colloquium on Automata, Languages and Programming (ICALP)}, keywords = {sys:relevantfor:ae andreasgoebel icalp leonschiller maximiliankatzmann tobiasfriedrich year2023}, pages = {62:1–62:13}, title = {Cliques in High-Dimensional Geometric Inhomogeneous Random Graphs}, year = 2023 }

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
[ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
 
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.
@inproceedings{bilo2023faulttolerant, abstract = {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.}, author = {Bilò, Davide and Choudhary, Keerti and Cohen, Sarel and Friedrich, Tobias and Krogmann, Simon and Schirneck, Martin}, booktitle = {International Colloquium on Automata, Languages and Programming (ICALP)}, keywords = {davidebilo icalp keertichoudhary martinschirneck sarelcohen simonkrogmann tobiasfriedrich year2023}, pages = {24:1-24:20}, title = {Fault-Tolerant ST-Diameter Oracles}, year = 2023 }

2022 [ nach oben ]

Friedrich, Tobias; Gawendowicz, Hans; Lenzner, Pascal; Melnichenko, Anna Social Distancing Network CreationInternational Colloquium on Automata, Languages and Programming (ICALP) 2022: 62:1–62:21
[ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
 
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.~[PODC 2003] where agents aim at being as central as possible in the created network. Thus, our work is in-line with studies that compare minimization problems with their maximization versions. We look at two variants of network creation governed by social distancing. In the first variant, there are no restrictions on the connections being formed. We characterize optimal and equilibrium networks, and we derive asymptotically tight bounds on the Price of Anarchy and Price of Stability. The second variant is the model's generalization that allows restrictions on the connections that can be formed. 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, compared the well-studied inverse models, under social distancing the agents' selfish behavior has a significantly stronger impact on the quality of the equilibria, i.e., allowing socially much worse stable states.
@inproceedings{friedrich2022social, abstract = {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. [PODC 2003] where agents aim at being as central as possible in the created network. Thus, our work is in-line with studies that compare minimization problems with their maximization versions. We look at two variants of network creation governed by social distancing. In the first variant, there are no restrictions on the connections being formed. We characterize optimal and equilibrium networks, and we derive asymptotically tight bounds on the Price of Anarchy and Price of Stability. The second variant is the model's generalization that allows restrictions on the connections that can be formed. 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, compared the well-studied inverse models, under social distancing the agents' selfish behavior has a significantly stronger impact on the quality of the equilibria, i.e., allowing socially much worse stable states.}, author = {Friedrich, Tobias and Gawendowicz, Hans and Lenzner, Pascal and Melnichenko, Anna}, booktitle = {International Colloquium on Automata, Languages and Programming (ICALP)}, keywords = {annamelnichenko hansgawendowicz icalp pascallenzner tobiasfriedrich year2022}, pages = {62:1–62:21}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum f{{ü}}r Informatik}, title = {Social Distancing Network Creation}, volume = {LIPIcs:229}, year = 2022 }

Bilò, Davide; Choudhary, Keerti; Cohen, Sarel; Friedrich, Tobias; Schirneck, Martin Deterministic Sensitivity Oracles for Diameter, Eccentricities and All Pairs DistancesInternational Colloquium on Automata, Languages and Programming (ICALP) 2022: 68:1–68:19
[ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
 
We construct data structures for extremal and pairwise distances in directed graphs in the presence of transient edge failures. Henzinger et al. (ITCS 2017) initiated the study of fault-tolerant (sensitivity) oracles for the diameter and vertex eccentricities. We extend this with a special focus on space efficiency. We present several new data structures, among them the first fault-tolerant eccentricity oracle for dual failures in subcubic space. We further prove lower bounds that show limits to approximation vs. space and diameter vs. space trade-offs for fault-tolerant oracles. They highlight key differences between data structures for undirected and directed graphs. Initially, our oracles are randomized leaning on a sampling technique frequently used in sensitivity analysis. Building on the work of Alon, Chechik, and Cohen (ICALP 2019) as well as Karthik and Parter (SODA 2021), we develop a hierarchical framework to derandomize fault-tolerant data structures. We first apply it to our own diameter/eccentricity oracles and then show its usefulness by derandomizing algorithms from the literature: the distance sensitivity oracle of Ren (JCSS 2022) and the Single-Source Replacement Path algorithm of Chechik and Magen (ICALP 2020).
@inproceedings{bilo2022deterministic, abstract = {We construct data structures for extremal and pairwise distances in directed graphs in the presence of transient edge failures. Henzinger et al. (ITCS 2017) initiated the study of fault-tolerant (sensitivity) oracles for the diameter and vertex eccentricities. We extend this with a special focus on space efficiency. We present several new data structures, among them the first fault-tolerant eccentricity oracle for dual failures in subcubic space. We further prove lower bounds that show limits to approximation vs. space and diameter vs. space trade-offs for fault-tolerant oracles. They highlight key differences between data structures for undirected and directed graphs. Initially, our oracles are randomized leaning on a sampling technique frequently used in sensitivity analysis. Building on the work of Alon, Chechik, and Cohen (ICALP 2019) as well as Karthik and Parter (SODA 2021), we develop a hierarchical framework to derandomize fault-tolerant data structures. We first apply it to our own diameter/eccentricity oracles and then show its usefulness by derandomizing algorithms from the literature: the distance sensitivity oracle of Ren (JCSS 2022) and the Single-Source Replacement Path algorithm of Chechik and Magen (ICALP 2020).}, author = {Bilò, Davide and Choudhary, Keerti and Cohen, Sarel and Friedrich, Tobias and Schirneck, Martin}, booktitle = {International Colloquium on Automata, Languages and Programming (ICALP)}, keywords = {davidebilo icalp keertichoudhary martinschirneck sarelcohen tobiasfriedrich year2022}, pages = {68:1-68:19}, title = {Deterministic Sensitivity Oracles for Diameter, Eccentricities and All Pairs Distances}, year = 2022 }

2021 [ nach oben ]

Lagodzinski, J. A. Gregor; Göbel, Andreas; Casel, Katrin; Friedrich, Tobias On Counting (Quantum-)Graph Homomorphisms in Finite Fields of Prime OrderInternational Colloquium on Automata, Languages and Programming (ICALP) 2021: 91:1–91:15
[ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
 
We study the problem of counting the number of homomorphisms from an input graph \(G\) to a fixed (quantum) graph \(\bar{H}\) in any finite field of prime order \(\mathbb{Z}_p\). The subproblem with graph \(H\) was introduced by Faben and Jerrum [ToC'15] and its complexity is still uncharacterised despite active research, e.g. the very recent work of Focke, Goldberg, Roth, and Zivný [SODA'21]. Our contribution is threefold. First, we introduce the study of quantum graphs to the study of modular counting homomorphisms. We show that the complexity for a quantum graph \(\bar{H}\) collapses to the complexity criteria found at dimension 1: graphs. Second, in order to prove cases of intractability we establish a further reduction to the study of bipartite graphs. Lastly, we establish a dichotomy for all bipartite \( (K_3,3 \ e, domino) \)-free graphs by a thorough structural study incorporating both local and global arguments. This result subsumes all results on bipartite graphs known for all prime moduli and extends them significantly. Even for the subproblem with \(p=2\) this establishes new results.
@inproceedings{lagodzinski2021counting, abstract = {We study the problem of counting the number of homomorphisms from an input graph \(G\) to a fixed (quantum) graph \(\bar{H}\) in any finite field of prime order \(\mathbb{Z}_p\). The subproblem with graph \(H\) was introduced by Faben and Jerrum [ToC'15] and its complexity is still uncharacterised despite active research, e.g. the very recent work of Focke, Goldberg, Roth, and Zivný [SODA'21]. Our contribution is threefold. First, we introduce the study of quantum graphs to the study of modular counting homomorphisms. We show that the complexity for a quantum graph \(\bar{H}\) collapses to the complexity criteria found at dimension 1: graphs. Second, in order to prove cases of intractability we establish a further reduction to the study of bipartite graphs. Lastly, we establish a dichotomy for all bipartite \( (K_{3,3} \ {e}, domino) \)-free graphs by a thorough structural study incorporating both local and global arguments. This result subsumes all results on bipartite graphs known for all prime moduli and extends them significantly. Even for the subproblem with \(p=2\) this establishes new results.}, author = {Lagodzinski, J. A. Gregor and Göbel, Andreas and Casel, Katrin and Friedrich, Tobias}, booktitle = {International Colloquium on Automata, Languages and Programming (ICALP)}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, keywords = {andreasgoebel gregorlagodzinski icalp katrincasel tobiasfriedrich year2021}, pages = {91:1–91:15}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum f{{ü}}r Informatik}, series = {LIPIcs}, title = {On Counting (Quantum-)Graph Homomorphisms in Finite Fields of Prime Order}, volume = 198, year = 2021 }

Friedrich, Tobias; Göbel, Andreas; Krejca, Martin S.; Pappik, Marcus A spectral independence view on hard spheres via block dynamicsInternational Colloquium on Automata, Languages and Programming (ICALP) 2021: 66:1–66:15
[ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
 
The hard-sphere model is one of the most extensively studied models in statistical physics. It describes the continuous distribution of spherical particles, governed by hard-core interactions. An important quantity of this model is the normalizing factor of this distribution, called the partition function. We propose a Markov chain Monte Carlo algorithm for approximating the grand-canonical partition function of the hard-sphere model in \(d\) dimensions. Up to a fugacity of \( \lambda < \text{e}/2^d\), the runtime of our algorithm is polynomial in the volume of the system. This covers the entire known real-valued regime for the uniqueness of the Gibbs measure. Key to our approach is to define a discretization that closely approximates the partition function of the continuous model. This results in a discrete hard-core instance that is exponential in the size of the initial hard-sphere model. Our approximation bound follows directly from the correlation decay threshold of an infinite regular tree with degree equal to the maximum degree of our discretization. To cope with the exponential blow-up of the discrete instance we use clique dynamics, a Markov chain that was recently introduced in the setting of abstract polymer models. We prove rapid mixing of clique dynamics up to the tree threshold of the univariate hard-core model. This is achieved by relating clique dynamics to block dynamics and adapting the spectral expansion method, which was recently used to bound the mixing time of Glauber dynamics within the same parameter regime.
@inproceedings{fgkp-asivohsvbd-2021, abstract = {The hard-sphere model is one of the most extensively studied models in statistical physics. It describes the continuous distribution of spherical particles, governed by hard-core interactions. An important quantity of this model is the normalizing factor of this distribution, called the partition function. We propose a Markov chain Monte Carlo algorithm for approximating the grand-canonical partition function of the hard-sphere model in \(d\) dimensions. Up to a fugacity of \( \lambda < \text{e}/2^d\), the runtime of our algorithm is polynomial in the volume of the system. This covers the entire known real-valued regime for the uniqueness of the Gibbs measure. Key to our approach is to define a discretization that closely approximates the partition function of the continuous model. This results in a discrete hard-core instance that is exponential in the size of the initial hard-sphere model. Our approximation bound follows directly from the correlation decay threshold of an infinite regular tree with degree equal to the maximum degree of our discretization. To cope with the exponential blow-up of the discrete instance we use clique dynamics, a Markov chain that was recently introduced in the setting of abstract polymer models. We prove rapid mixing of clique dynamics up to the tree threshold of the univariate hard-core model. This is achieved by relating clique dynamics to block dynamics and adapting the spectral expansion method, which was recently used to bound the mixing time of Glauber dynamics within the same parameter regime.}, author = {Friedrich, Tobias and Göbel, Andreas and Krejca, Martin S. and Pappik, Marcus}, booktitle = {International Colloquium on Automata, Languages and Programming (ICALP)}, keywords = {andreasgoebel icalp marcuspappik martinskrejca tobiasfriedrich year2021}, pages = {66:1-66:15}, title = {A spectral independence view on hard spheres via block dynamics}, volume = 198, year = 2021 }

2019 [ nach oben ]

Alon, Noga; Chechik, Shiri; Cohen, Sarel Deterministic Combinatorial Replacement Paths and Distance Sensitivity OraclesInternational Colloquium on Automata, Languages and Programming (ICALP) 2019: 12:1–12:14
[ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
 
In this work we derandomize two central results in graph algorithms, replacement paths and distance sensitivity oracles (DSOs) matching in both cases the running time of the randomized algorithms. For the replacement paths problem, let \(G = (V,E)\) be a directed unweighted graph with \(n\) vertices and m edges and let \(P\) be a shortest path from \(s\) to \(t\) in \(G\). The replacement paths problem is to find for every edge \(e\) in \(P\) the shortest path from \(s\) to \(t\) avoiding \(e\). Roditty and Zwick [ICALP 2005] obtained a randomized algorithm with running time of \(\mathcal{O(m \sqrt{n})\). Here we provide the first deterministic algorithm for this problem, with the same \(\mathcal{O(m \sqrt{n})\) time. Due to matching conditional lower bounds of Williams et al. [FOCS 2010], our deterministic combinatorial algorithm for the replacement paths problem is optimal up to polylogarithmic factors (unless the long standing bound of \(\mathcal{O(mn)\) for the combinatorial boolean matrix multiplication can be improved). This also implies a deterministic algorithm for the second simple shortest path problem in \(\mathcal{O(m \sqrt{n})\) time, and a deterministic algorithm for the \(k\)-simple shortest paths problem in \(\mathcal{O(k m sqrt{n})\) time (for any integer constant \(k > 0\)). For the problem of distance sensitivity oracles, let \(G = (V,E)\) be a directed graph with real-edge weights. An \(f\)-Sensitivity Distance Oracle (\(f\)-DSO) gets as input the graph \(G=(V,E)\) and a parameter \(f\), preprocesses it into a data-structure, such that given a query \((s,t,F)\) with \(s,t \in V\) and \(F \subseteq E \cup V\), \(|F| <=f\) being a set of at most \(f\) edges or vertices (failures), the query algorithm efficiently computes the distance from \(s\) to \(t\) in the graph \(G \setminus F\) (i.e., the distance from \(s\) to \(t\) in the graph \(G\) after removing from it the failing edges and vertices \(F\)). For weighted graphs with real edge weights, Weimann and Yuster [FOCS 2010] presented several randomized \(f\)-DSOs. In particular, they presented a combinatorial \(f\)-DSO with \(\mathcal{O(mn^{4-\alpha})\) preprocessing time and subquadratic \(\mathcal{O(n^2-2(1-\alpha)/f})\) query time, giving a tradeoff between preprocessing and query time for every value of \(0 < \alpha < 1\). We derandomize this result and present a combinatorial deterministic \(f\)-DSO with the same asymptotic preprocessing and query time.
@inproceedings{alon2019deterministic, abstract = {In this work we derandomize two central results in graph algorithms, replacement paths and distance sensitivity oracles (DSOs) matching in both cases the running time of the randomized algorithms. For the replacement paths problem, let \(G = (V,E)\) be a directed unweighted graph with \(n\) vertices and m edges and let \(P\) be a shortest path from \(s\) to \(t\) in \(G\). The replacement paths problem is to find for every edge \(e\) in \(P\) the shortest path from \(s\) to \(t\) avoiding \(e\). Roditty and Zwick [ICALP 2005] obtained a randomized algorithm with running time of \(\mathcal{O}(m \sqrt{n})\). Here we provide the first deterministic algorithm for this problem, with the same \(\mathcal{O}(m \sqrt{n})\) time. Due to matching conditional lower bounds of Williams et al. [FOCS 2010], our deterministic combinatorial algorithm for the replacement paths problem is optimal up to polylogarithmic factors (unless the long standing bound of \(\mathcal{O}(mn)\) for the combinatorial boolean matrix multiplication can be improved). This also implies a deterministic algorithm for the second simple shortest path problem in \(\mathcal{O}(m \sqrt{n})\) time, and a deterministic algorithm for the \(k\)-simple shortest paths problem in \(\mathcal{O}(k m sqrt{n})\) time (for any integer constant \(k > 0\)). For the problem of distance sensitivity oracles, let \(G = (V,E)\) be a directed graph with real-edge weights. An \(f\)-Sensitivity Distance Oracle (\(f\)-DSO) gets as input the graph \(G=(V,E)\) and a parameter \(f\), preprocesses it into a data-structure, such that given a query \((s,t,F)\) with \(s,t \in V\) and \(F \subseteq E \cup V\), \(|F| <=f\) being a set of at most \(f\) edges or vertices (failures), the query algorithm efficiently computes the distance from \(s\) to \(t\) in the graph \(G \setminus F\) (i.e., the distance from \(s\) to \(t\) in the graph \(G\) after removing from it the failing edges and vertices \(F\)). For weighted graphs with real edge weights, Weimann and Yuster [FOCS 2010] presented several randomized \(f\)-DSOs. In particular, they presented a combinatorial \(f\)-DSO with \(\mathcal{O}(mn^{4-\alpha})\) preprocessing time and subquadratic \(\mathcal{O}(n^{2-2(1-\alpha)/f})\) query time, giving a tradeoff between preprocessing and query time for every value of \(0 < \alpha < 1\). We derandomize this result and present a combinatorial deterministic \(f\)-DSO with the same asymptotic preprocessing and query time.}, author = {Alon, Noga and Chechik, Shiri and Cohen, Sarel}, booktitle = {International Colloquium on Automata, Languages and Programming (ICALP)}, keywords = {sys:relevantfor:ae icalp nogaalon sarelcohen shirichechik year2019}, pages = {12:1-12:14}, title = {Deterministic Combinatorial Replacement Paths and Distance Sensitivity Oracles}, year = 2019 }

Friedrich, Tobias; Rothenberger, Ralf The Satisfiability Threshold for Non-Uniform Random 2-SATInternational Colloquium on Automata, Languages and Programming (ICALP) 2019: 61:1–61:14
[ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
 
Propositional satisfiability (SAT) is one of the most fundamental problems in computer science. Its worst-case hardness lies at the core of computational complexity theory, for example in the form of NP-hardness and the (Strong) Exponential Time Hypothesis. In practice however, SAT instances can often be solved efficiently. This contradicting behavior has spawned interest in the average-case analysis of SAT and has triggered the development of sophisticated rigorous and non-rigorous techniques for analyzing random structures. Despite a long line of research and substantial progress, most theoretical work on random SAT assumes a uniform distribution on the variables. In contrast, real-world instances often exhibit large fluctuations in variable occurrence. This can be modeled by a non-uniform distribution of the variables, which can result in distributions closer to industrial SAT instances. We study satisfiability thresholds of non-uniform random 2-SAT with n variables and m clauses and with an arbitrary probability distribution \((p_i)_{i \in [n]}\) with \(p_1 \ge p_2 \ge \dots \ge p_n > 0\) over the n variables. We show for \(p_1^2 = \Theta(\sum_{i=1^n p_i^2)\) that the asymptotic satisfiability threshold is at \(m = \Theta((1− \sum_{i=1^n p_i^2) / \)\((p_1 (\sum_{i=2^n p_i^2)^{1/2}))\) and that it is coarse. For \(p_1^2 = o( \sum_{i=1^n p_i^2)\) we show that there is a sharp satisfiability threshold at \(m = (\sum_{i=1^n p_i^2)^{−1}\). This result generalizes the seminal works by Chvatal and Reed [FOCS 1992] and by Goerdt [JCSS 1996].
@inproceedings{friedrich2019satisfiabilityICALP, abstract = {Propositional satisfiability (SAT) is one of the most fundamental problems in computer science. Its worst-case hardness lies at the core of computational complexity theory, for example in the form of NP-hardness and the (Strong) Exponential Time Hypothesis. In practice however, SAT instances can often be solved efficiently. This contradicting behavior has spawned interest in the average-case analysis of SAT and has triggered the development of sophisticated rigorous and non-rigorous techniques for analyzing random structures. Despite a long line of research and substantial progress, most theoretical work on random SAT assumes a uniform distribution on the variables. In contrast, real-world instances often exhibit large fluctuations in variable occurrence. This can be modeled by a non-uniform distribution of the variables, which can result in distributions closer to industrial SAT instances. We study satisfiability thresholds of non-uniform random 2-SAT with n variables and m clauses and with an arbitrary probability distribution \((p_i)_{i \in [n]}\) with \(p_1 \ge p_2 \ge \dots \ge p_n > 0\) over the n variables. We show for \(p_1^2 = \Theta(\sum_{i=1}^n p_i^2)\) that the asymptotic satisfiability threshold is at \(m = \Theta((1− \sum_{i=1}^n p_i^2) / \)\((p_1 (\sum_{i=2}^n p_i^2)^{1/2}))\) and that it is coarse. For \(p_1^2 = o( \sum_{i=1}^n p_i^2)\) we show that there is a sharp satisfiability threshold at \(m = (\sum_{i=1}^n p_i^2)^{−1}\). This result generalizes the seminal works by Chvatal and Reed [FOCS 1992] and by Goerdt [JCSS 1996].}, author = {Friedrich, Tobias and Rothenberger, Ralf}, booktitle = {International Colloquium on Automata, Languages and Programming (ICALP)}, keywords = {icalp ralfrothenberger scalefree_sat tobiasfriedrich year2019}, pages = {61:1-61:14}, title = {The Satisfiability Threshold for Non-Uniform Random 2-SAT}, year = 2019 }

Casel, Katrin; Day, Joel D.; Fleischmann, Pamela; Kociumaka, Tomasz; Manea, Florin; Schmid, Markus L. Graph and String Parameters: Connections Between Pathwidth, Cutwidth and the Locality NumberInternational Colloquium on Automata, Languages and Programming (ICALP) 2019: 109:1–109:16
[ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
 
We investigate the locality number, a recently introduced structural parameter for strings (with applications in pattern matching with variables), and its connection to two important graph-parameters, cutwidth and pathwidth. These connections allow us to show that computing the locality number is NP-hard but fixed parameter tractable (when the locality number or the alphabet size is treated as a parameter), and can be approximated with ratio O( \( \sqrt{ \log opt \log n \) ). As a by-product, we also relate cutwidth via the locality number to pathwidth, which is of independent interest, since it improves the currently best known approximation algorithm for cutwidth. In addition to these main results, we also consider the possibility of greedy-based approximation algorithms for the locality number.
@inproceedings{casel2019graph, abstract = {We investigate the locality number, a recently introduced structural parameter for strings (with applications in pattern matching with variables), and its connection to two important graph-parameters, cutwidth and pathwidth. These connections allow us to show that computing the locality number is NP-hard but fixed parameter tractable (when the locality number or the alphabet size is treated as a parameter), and can be approximated with ratio O( \( \sqrt{ \log opt} \log n \) ). As a by-product, we also relate cutwidth via the locality number to pathwidth, which is of independent interest, since it improves the currently best known approximation algorithm for cutwidth. In addition to these main results, we also consider the possibility of greedy-based approximation algorithms for the locality number.}, author = {Casel, Katrin and Day, Joel D. and Fleischmann, Pamela and Kociumaka, Tomasz and Manea, Florin and Schmid, Markus L.}, booktitle = {International Colloquium on Automata, Languages and Programming (ICALP)}, keywords = {icalp katrincasel year2019}, pages = {109:1-109:16}, title = {Graph and String Parameters: Connections Between Pathwidth, Cutwidth and the Locality Number}, year = 2019 }

2018 [ nach oben ]

Bläsius, Thomas; Freiberger, Cedric; Friedrich, Tobias; Katzmann, Maximilian; Montenegro-Retana, Felix; Thieffry, Marianne Efficient Shortest Paths in Scale-Free Networks with Underlying Hyperbolic GeometryInternational Colloquium on Automata, Languages, and Programming (ICALP) 2018: 20:1–20:14
[ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
 
A common way to accelerate shortest path algorithms on graphs is the use of a bidirectional search, which simultaneously explores the graph from the start and the destination. It has been observed recently that this strategy performs particularly well on scale-free real-world networks. Such networks typically have a heterogeneous degree distribution (e.g., a power-law distribution) and high clustering (i.e., vertices with a common neighbor are likely to be connected themselves). These two properties can be obtained by assuming an underlying hyperbolic geometry. To explain the observed behavior of the bidirectional search, we analyze its running time on hyperbolic random graphs and prove that it is \(\tilde{O}(n\)\(^{2 - 1/ \alpha}+\)\(n^{1/(2\alpha)}\)\(+ \delta_{\max})\) with high probability, where \(\alpha\)\(\in\)\((0.5, 1)\) controls the power-law exponent of the degree distribution, and \(\delta_{\max}\) is the maximum degree. This bound is sublinear, improving the obvious worst-case linear bound. Although our analysis depends on the underlying geometry, the algorithm itself is oblivious to it.
@inproceedings{blasius2018efficient, abstract = {A common way to accelerate shortest path algorithms on graphs is the use of a bidirectional search, which simultaneously explores the graph from the start and the destination. It has been observed recently that this strategy performs particularly well on scale-free real-world networks. Such networks typically have a heterogeneous degree distribution (e.g., a power-law distribution) and high clustering (i.e., vertices with a common neighbor are likely to be connected themselves). These two properties can be obtained by assuming an underlying hyperbolic geometry. To explain the observed behavior of the bidirectional search, we analyze its running time on hyperbolic random graphs and prove that it is \(\tilde{O}(n\)\(^{2 - 1/ \alpha}+\)\(n^{1/(2\alpha)}\)\(+ \delta_{\max})\) with high probability, where \(\alpha\)\(\in\)\((0.5, 1)\) controls the power-law exponent of the degree distribution, and \(\delta_{\max}\) is the maximum degree. This bound is sublinear, improving the obvious worst-case linear bound. Although our analysis depends on the underlying geometry, the algorithm itself is oblivious to it.}, author = {Bläsius, Thomas and Freiberger, Cedric and Friedrich, Tobias and Katzmann, Maximilian and Montenegro-Retana, Felix and Thieffry, Marianne}, booktitle = {International Colloquium on Automata, Languages, and Programming (ICALP)}, keywords = {cedricfreiberger felixmontenegroretana hyperbolic_analysis icalp mariannethieffry maximiliankatzmann thomasblaesius tobiasfriedrich year2018}, pages = {20:1-20:14}, title = {Efficient Shortest Paths in Scale-Free Networks with Underlying Hyperbolic Geometry}, volume = 107, year = 2018 }

Issac, Davis; Chandran, L. Sunil; Cheung, Yuen Kueng Spanning tree congestion and computation of gyori lovasz partitionInternational Colloquium on Automata, Languages, and Programming (ICALP) 2018: 1–14
[ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
 
We study a natural problem in graph sparsification, the Spanning Tree Congestion (STC) problem. Informally, it seeks a spanning tree with no tree-edge routing too many of the original edges. For any general connected graph with n vertices and m edges, we show that its STC is at most \( \mathcal{O(\sqrt{mn}) \), which is asymptotically optimal since we also demonstrate graphs with STC at least \( \Omega(\sqrt{mn}) \). We present a polynomial-time algorithm which computes a spanning tree with congestion \( \mathcal{O(\sqrt{mn}) \log n \) \( \mathcal{O(\sqrt{mn} \log n ) \). We also present another algorithm for computing a spanning tree with congestion \( \mathcal{O(\sqrt{mn}) \); this algorithm runs in sub-exponential time when \(m = \omega(n \log^2 n ) \). For achieving the above results, an important intermediate theorem is generalized Györi-Lovász theorem. Chen et al. [Jiangzhuo Chen et al., 2007] gave a non-constructive proof. We give the first elementary and constructive proof with a local search algorithm of running time \( \mathcal{O^*( 4^n ) \). We discuss some consequences of the theorem concerning graph partitioning, which might be of independent interest. We also show that for any graph which satisfies certain expanding properties, its STC is at most \( \mathcal{O^*(n) \), and a corresponding spanning tree can be computed in polynomial time. We then use this to show that a random graph has STC \( \Theta(n) \) with high probability.
@inproceedings{icc-stccglp-2018, abstract = {We study a natural problem in graph sparsification, the Spanning Tree Congestion (STC) problem. Informally, it seeks a spanning tree with no tree-edge routing too many of the original edges. For any general connected graph with n vertices and m edges, we show that its STC is at most \( \mathcal{O}(\sqrt{mn}) \), which is asymptotically optimal since we also demonstrate graphs with STC at least \( \Omega(\sqrt{mn}) \). We present a polynomial-time algorithm which computes a spanning tree with congestion \( \mathcal{O}(\sqrt{mn}) \log n \) \( \mathcal{O}(\sqrt{mn} \log n ) \). We also present another algorithm for computing a spanning tree with congestion \( \mathcal{O}(\sqrt{mn}) \); this algorithm runs in sub-exponential time when \(m = \omega(n \log^2 n ) \). For achieving the above results, an important intermediate theorem is generalized Györi-Lovász theorem. Chen et al. [Jiangzhuo Chen et al., 2007] gave a non-constructive proof. We give the first elementary and constructive proof with a local search algorithm of running time \( \mathcal{O}^*( 4^n ) \). We discuss some consequences of the theorem concerning graph partitioning, which might be of independent interest. We also show that for any graph which satisfies certain expanding properties, its STC is at most \( \mathcal{O}^*(n) \), and a corresponding spanning tree can be computed in polynomial time. We then use this to show that a random graph has STC \( \Theta(n) \) with high probability.}, author = {Issac, Davis and Chandran, L. Sunil and Cheung, Yuen Kueng}, booktitle = {International Colloquium on Automata, Languages, and Programming (ICALP)}, keywords = {davisissac icalp lsunilchandran year2018 yuenkuencheung}, pages = {1-14}, title = {Spanning tree congestion and computation of gyori lovasz partition}, year = 2018 }

Arar, Moab; Chechik, Shiri; Cohen, Sarel; Stein, Cliff; Wajc, David Dynamic Matching: Reducing Integral Algorithms to Approximately-Maximal Fractional AlgorithmsInternational Colloquium on Automata, Languages and Programming (ICALP) 2018: 7:1–7:16
[ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
 
We present a simple randomized reduction from fully-dynamic integral matching algorithms to fully-dynamic "approximately-maximal" fractional matching algorithms. Applying this reduction to the recent fractional matching algorithm of Bhattacharya, Henzinger, and Nanongkai (SODA 2017), we obtain a novel result for the integral problem. Specifically, our main result is a randomized fully-dynamic \((2+\varepsilon)\)-approximate integral matching algorithm with small polylog worst-case update time. For the \((2+\varepsilon)\)-approximation regime only a fractional fully-dynamic \((2+\varepsilon)\)-matching algorithm with worst-case polylog update time was previously known, due to Bhattacharya et al. (SODA 2017). Our algorithm is the first algorithm that maintains approximate matchings with worst-case update time better than polynomial, for any constant approximation ratio. As a consequence, we also obtain the first constant-approximate worst-case polylogarithmic update time maximum weight matching algorithm.
@inproceedings{arar2018dynamic, abstract = {We present a simple randomized reduction from fully-dynamic integral matching algorithms to fully-dynamic "approximately-maximal" fractional matching algorithms. Applying this reduction to the recent fractional matching algorithm of Bhattacharya, Henzinger, and Nanongkai (SODA 2017), we obtain a novel result for the integral problem. Specifically, our main result is a randomized fully-dynamic \((2+\varepsilon)\)-approximate integral matching algorithm with small polylog worst-case update time. For the \((2+\varepsilon)\)-approximation regime only a fractional fully-dynamic \((2+\varepsilon)\)-matching algorithm with worst-case polylog update time was previously known, due to Bhattacharya et al. (SODA 2017). Our algorithm is the first algorithm that maintains approximate matchings with worst-case update time better than polynomial, for any constant approximation ratio. As a consequence, we also obtain the first constant-approximate worst-case polylogarithmic update time maximum weight matching algorithm.}, author = {Arar, Moab and Chechik, Shiri and Cohen, Sarel and Stein, Cliff and Wajc, David}, booktitle = {International Colloquium on Automata, Languages and Programming (ICALP)}, keywords = {sys:relevantfor:ae cliffstein davidwajc icalp moabarar sarelcohen shirichechik year2018}, pages = {7:1-7:16}, title = {Dynamic Matching: Reducing Integral Algorithms to Approximately-Maximal Fractional Algorithms}, year = 2018 }

2017 [ nach oben ]

Michail, Othon; Skretas, George; Spirakis, G. Paul On the Transformation Capability of Feasible Mechanisms for Programmable MatterInternational Colloquium on Automata, Languages and Programming (ICALP) 2017: 136:1–136:15
[ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
 
In this work, we study theoretical models of programmable matter systems. The systems under consideration consist of spherical modules, kept together by magnetic forces and able to perform two minimal mechanical operations (or movements): rotate around a neighbor and slide over a line. In terms of modeling, there are n nodes arranged in a 2-dimensional grid and forming some initial shape. The goal is for the initial shape A to transform to some target shape B by a sequence of movements. Most of the paper focuses on transformability questions, meaning whether it is in principle feasible to transform a given shape to another. We first consider the case in which only rotation is available to the nodes. Our main result is that deciding whether two given shapes A and B can be transformed to each other is in P. We then insist on rotation only and impose the restriction that the nodes must maintain global connectivity throughout the transformation. We prove that the corresponding transformability question is in PSPACE and study the problem of determining the minimum seeds that can make feasible otherwise infeasible transformations. Next we allow both rotations and slidings and prove universality: any two connected shapes A,B of the same number of nodes, can be transformed to each other without breaking connectivity. The worst-case number of movements of the generic strategy is \(\Theta(n^2)\). We improve this to \( \mathcal{O(n) \) parallel time, by a pipelining strategy, and prove optimality of both by matching lower bounds. We next turn our attention to distributed transformations. The nodes are now distributed processes able to perform communicate-compute-move rounds. We provide distributed algorithms for a general type of transformation.
@inproceedings{michail2017transformation, abstract = {In this work, we study theoretical models of programmable matter systems. The systems under consideration consist of spherical modules, kept together by magnetic forces and able to perform two minimal mechanical operations (or movements): rotate around a neighbor and slide over a line. In terms of modeling, there are n nodes arranged in a 2-dimensional grid and forming some initial shape. The goal is for the initial shape A to transform to some target shape B by a sequence of movements. Most of the paper focuses on transformability questions, meaning whether it is in principle feasible to transform a given shape to another. We first consider the case in which only rotation is available to the nodes. Our main result is that deciding whether two given shapes A and B can be transformed to each other is in P. We then insist on rotation only and impose the restriction that the nodes must maintain global connectivity throughout the transformation. We prove that the corresponding transformability question is in PSPACE and study the problem of determining the minimum seeds that can make feasible otherwise infeasible transformations. Next we allow both rotations and slidings and prove universality: any two connected shapes A,B of the same number of nodes, can be transformed to each other without breaking connectivity. The worst-case number of movements of the generic strategy is \(\Theta(n^2)\). We improve this to \( \mathcal{O}(n) \) parallel time, by a pipelining strategy, and prove optimality of both by matching lower bounds. We next turn our attention to distributed transformations. The nodes are now distributed processes able to perform communicate-compute-move rounds. We provide distributed algorithms for a general type of transformation.}, author = {Michail, Othon and Skretas, George and Spirakis, G. Paul}, booktitle = {International Colloquium on Automata, Languages and Programming (ICALP)}, keywords = {sys:relevantfor:ae georgeskretas icalp michailothon paulgspirakis year2017}, pages = {136:1&ndash;136:15}, title = {On the Transformation Capability of Feasible Mechanisms for Programmable Matter}, year = 2017 }

2016 [ nach oben ]

Casel, Katrin; Fernau, Henning; Gaspers, Serge; Gras, Benjamin; Schmid, Markus L. On the Complexity of Grammar-Based Compression over Fixed AlphabetsInternational Colloquium on Automata, Languages, and Programming (ICALP) 2016: 122:1–122:14
[ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
 
It is shown that the shortest-grammar problem remains NP-complete if the alphabet is fixed and has a size of at least 24 (which settles an open question). On the other hand, this problem can be solved in polynomial-time, if the number of nonterminals is bounded, which is shown by encoding the problem as a problem on graphs with interval structure. Furthermore, we present an \(O(3^n)\) exact exponential-time algorithm, based on dynamic programming. Similar results are also given for 1-level grammars, i.e., grammars for which only the start rule contains nonterminals on the right side (thus, investigating the impact of the "hierarchical depth" on the complexity of the shortest-grammar problem).
@inproceedings{DBLP:conf/icalp/CaselFGGS16, abstract = {It is shown that the shortest-grammar problem remains NP-complete if the alphabet is fixed and has a size of at least 24 (which settles an open question). On the other hand, this problem can be solved in polynomial-time, if the number of nonterminals is bounded, which is shown by encoding the problem as a problem on graphs with interval structure. Furthermore, we present an \(O(3^n)\) exact exponential-time algorithm, based on dynamic programming. Similar results are also given for 1-level grammars, i.e., grammars for which only the start rule contains nonterminals on the right side (thus, investigating the impact of the "hierarchical depth" on the complexity of the shortest-grammar problem).}, author = {Casel, Katrin and Fernau, Henning and Gaspers, Serge and Gras, Benjamin and Schmid, Markus L.}, booktitle = {International Colloquium on Automata, Languages, and Programming (ICALP)}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, keywords = {benjamingras henningfernau icalp katrincasel markuslschmid sergegaspers year2016}, pages = {122:1-122:14}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik}, series = {LIPIcs}, title = {On the Complexity of Grammar-Based Compression over Fixed Alphabets}, volume = 55, year = 2016 }

Galanis, Andreas; Göbel, Andreas; Goldberg, Leslie-Ann; Lapinskas, John; Richerby, David Amplifiers for the Moran ProcessInternational Colloquium on Automata, Languages and Programming (ICALP) 2016: 62:1–62:13
[ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
 
The Moran process, as studied by Lieberman, Hauert and Nowak, is a randomised algorithm modelling the spread of genetic mutations in populations. The algorithm runs on an underlying graph where individuals correspond to vertices. Initially, one vertex (chosen uniformly at random) possesses a mutation, with fitness \(r > 1\). All other individuals have fitness 1. During each step of the algorithm, an individual is chosen with probability proportional to its fitness, and its state (mutant or non-mutant) is passed on to an out-neighbour which is chosen uniformly at random. If the underlying graph is strongly connected then the algorithm will eventually reach fixation, in which all individuals are mutants, or extinction, in which no individuals are mutants. An infinite family of directed graphs is said to be strongly amplifying if, for every \(r > 1\), the extinction probability tends to 0 as the number of vertices increases. Strong amplification is a rather surprising property - it means that in such graphs, the fixation probability of a uniformly-placed initial mutant tends to 1 even though the initial mutant only has a fixed selective advantage of \(r > 1\) (independently of \(n\)). The name "strongly amplifying" comes from the fact that this selective advantage is "amplified". Strong amplifiers have received quite a bit of attention, and Lieberman et al. proposed two potentially strongly-amplifying families - superstars and metafunnels. Heuristic arguments have been published, arguing that there are infinite families of superstars that are strongly amplifying. The same has been claimed for metafunnels. We give the first rigorous proof that there is an infinite family of directed graphs that is strongly amplifying. We call the graphs in the family "megastars". When the algorithm is run on an n-vertex graph in this family, starting with a uniformly-chosen mutant, the extinction probability is roughly \(n^{-1/2}\) (up to logarithmic factors). We prove that all infinite families of superstars and metafunnels have larger extinction probabilities (as a function of \(n\)). Finally, we prove that our analysis of megastars is fairly tight - there is no infinite family of megastars such that the Moran algorithm gives a smaller extinction probability (up to logarithmic factors). Also, we provide a counterexample which clarifies the literature concerning the isothermal theorem of Lieberman et al. A full version [Galanis/Göbel/Goldberg/Lapinskas/Richerby, Preprint] containing detailed proofs is available at http://arxiv.org/abs/1512.05632. Theorem-numbering here matches the full version.
@inproceedings{DBLP:conf/icalp/Galanis0GLR16, abstract = {The Moran process, as studied by Lieberman, Hauert and Nowak, is a randomised algorithm modelling the spread of genetic mutations in populations. The algorithm runs on an underlying graph where individuals correspond to vertices. Initially, one vertex (chosen uniformly at random) possesses a mutation, with fitness \(r > 1\). All other individuals have fitness 1. During each step of the algorithm, an individual is chosen with probability proportional to its fitness, and its state (mutant or non-mutant) is passed on to an out-neighbour which is chosen uniformly at random. If the underlying graph is strongly connected then the algorithm will eventually reach fixation, in which all individuals are mutants, or extinction, in which no individuals are mutants. An infinite family of directed graphs is said to be strongly amplifying if, for every \(r > 1\), the extinction probability tends to 0 as the number of vertices increases. Strong amplification is a rather surprising property - it means that in such graphs, the fixation probability of a uniformly-placed initial mutant tends to 1 even though the initial mutant only has a fixed selective advantage of \(r > 1\) (independently of \(n\)). The name "strongly amplifying" comes from the fact that this selective advantage is "amplified". Strong amplifiers have received quite a bit of attention, and Lieberman et al. proposed two potentially strongly-amplifying families - superstars and metafunnels. Heuristic arguments have been published, arguing that there are infinite families of superstars that are strongly amplifying. The same has been claimed for metafunnels. We give the first rigorous proof that there is an infinite family of directed graphs that is strongly amplifying. We call the graphs in the family "megastars". When the algorithm is run on an n-vertex graph in this family, starting with a uniformly-chosen mutant, the extinction probability is roughly \(n^{-1/2}\) (up to logarithmic factors). We prove that all infinite families of superstars and metafunnels have larger extinction probabilities (as a function of \(n\)). Finally, we prove that our analysis of megastars is fairly tight - there is no infinite family of megastars such that the Moran algorithm gives a smaller extinction probability (up to logarithmic factors). Also, we provide a counterexample which clarifies the literature concerning the isothermal theorem of Lieberman et al. A full version [Galanis/Göbel/Goldberg/Lapinskas/Richerby, Preprint] containing detailed proofs is available at http://arxiv.org/abs/1512.05632. Theorem-numbering here matches the full version.}, author = {Galanis, Andreas and Göbel, Andreas and Goldberg, Leslie-Ann and Lapinskas, John and Richerby, David}, booktitle = {International Colloquium on Automata, Languages and Programming (ICALP)}, keywords = {andreasgoebel icalp year2016}, pages = {62:1-62:13}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik}, series = {LIPIcs}, title = {Amplifiers for the Moran Process}, volume = 55, year = 2016 }

2015 [ nach oben ]

Cseh, Ágnes; Huang, Chien-Chung; Kavitha, Telikepalli Popular Matchings with Two-Sided Preferences and One-Sided TiesInternational Colloquium on Automata, Languages and Programming (ICALP) 2015: 367–379
[ Abstract ] [ BibTeX ] [ URL ] [ DOI ]
 
We are given a bipartite graph \($G=(A \cup B, E)$\) where each vertex has a preference list ranking its neighbors: in particular, every \($a \in A$\) ranks its neighbors in a strict order of preference, whereas the preference lists of \($b \in B$\) may contain ties. A matching \($M$\) is popular if there is no matching \($M'$\) such that the number of vertices that prefer \($M'$\) to \($M$\) exceeds the number that prefer \($M$\) to \($M'$\). We show that the problem of deciding whether \($G$\) admits a popular matching or not is NP-hard. This is the case even when every \($b \in B$\) either has a strict preference list or puts all its neighbors into a single tie. In contrast, we show that the problem becomes polynomially solvable in the case when each \($b \in B$\) puts all its neighbors into a single tie. That is, all neighbors of \($b$\) are tied in \($b’s$\) list and and b desires to be matched to any of them. Our main result is an \($\mathcal{O(n^2)$\) algorithm (where \($n=|A \cup B|$\)) for the popular matching problem in this model. Note that this model is quite different from the model where vertices in \($B$\) have no preferences and do not care whether they are matched or not.
@inproceedings{cseh2015popular, abstract = {We are given a bipartite graph $G=(A \cup B, E)$ where each vertex has a preference list ranking its neighbors: in particular, every $a \in A$ ranks its neighbors in a strict order of preference, whereas the preference lists of $b \in B$ may contain ties. A matching $M$ is popular if there is no matching $M'$ such that the number of vertices that prefer $M'$ to $M$ exceeds the number that prefer $M$ to $M'$. We show that the problem of deciding whether $G$ admits a popular matching or not is NP-hard. This is the case even when every $b \in B$ either has a strict preference list or puts all its neighbors into a single tie. In contrast, we show that the problem becomes polynomially solvable in the case when each $b \in B$ puts all its neighbors into a single tie. That is, all neighbors of $b$ are tied in $b’s$ list and and b desires to be matched to any of them. Our main result is an $\mathcal{O}(n^2)$ algorithm (where $n=|A \cup B|$) for the popular matching problem in this model. Note that this model is quite different from the model where vertices in $B$ have no preferences and do not care whether they are matched or not.}, author = {Cseh, Ágnes and Huang, Chien-Chung and Kavitha, Telikepalli}, booktitle = {International Colloquium on Automata, Languages and Programming (ICALP)}, keywords = {sys:relevantfor:ae agnescseh chien-chunghuang icalp telikepallikavitha year2015}, pages = {367-379}, title = {Popular Matchings with Two-Sided Preferences and One-Sided Ties}, year = 2015 }

Göbel, Andreas; Goldberg, Leslie Ann; Richerby, David Counting Homomorphisms to Square-Free Graphs, Modulo 2International Colloquium on Automata, Languages, and Programming (ICALP) 2015: 642–653
[ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
 
We study the problem \( \oplus \)HomsToH of counting, modulo 2, the homomorphisms from an input graph to a fixed undirected graph \(H\). A characteristic feature of modular counting is that cancellations make wider classes of instances tractable than is the case for exact (nonmodular) counting; thus, subtle dichotomy theorems can arise. We show the following dichotomy: for any \(H\) that contains no 4-cycles, \(\oplus\)HomsToH is either in polynomial time or is \(\oplus\)P-complete. This partially confirms a conjecture of Faben and Jerrum that was previously only known to hold for trees and for a restricted class of tree-width-2 graphs called cactus graphs. We confirm the conjecture for a rich class of graphs, including graphs of unbounded tree-width. In particular, we focus on square-free graphs, which are graphs without 4-cycles. These graphs arise frequently in combinatorics, for example, in connection with the strong perfect graph theorem and in certain graph algorithms. Previous dichotomy theorems required the graph to be tree-like so that tree-like decompositions could be exploited in the proof. We prove the conjecture for a much richer class of graphs by adopting a much more general approach.
@inproceedings{DBLP:conf/icalp/0001GR15, abstract = {We study the problem \( \oplus \)HomsToH of counting, modulo 2, the homomorphisms from an input graph to a fixed undirected graph \(H\). A characteristic feature of modular counting is that cancellations make wider classes of instances tractable than is the case for exact (nonmodular) counting; thus, subtle dichotomy theorems can arise. We show the following dichotomy: for any \(H\) that contains no 4-cycles, \(\oplus\)HomsToH is either in polynomial time or is \(\oplus\)P-complete. This partially confirms a conjecture of Faben and Jerrum that was previously only known to hold for trees and for a restricted class of tree-width-2 graphs called cactus graphs. We confirm the conjecture for a rich class of graphs, including graphs of unbounded tree-width. In particular, we focus on square-free graphs, which are graphs without 4-cycles. These graphs arise frequently in combinatorics, for example, in connection with the strong perfect graph theorem and in certain graph algorithms. Previous dichotomy theorems required the graph to be tree-like so that tree-like decompositions could be exploited in the proof. We prove the conjecture for a much richer class of graphs by adopting a much more general approach.}, author = {Göbel, Andreas and Goldberg, Leslie Ann and Richerby, David}, booktitle = {International Colloquium on Automata, Languages, and Programming (ICALP)}, keywords = {andreasgoebel icalp notaccepted year2015}, pages = {642-653}, title = {Counting Homomorphisms to Square-Free Graphs, Modulo 2}, year = 2015 }

Bringmann, Karl; Friedrich, Tobias; Hoefer, Martin; Rothenberger, Ralf; Sauerwald, Thomas Ultra-Fast Load Balancing on Scale-Free NetworksInternational Colloquium on Automata, Languages and Programming (ICALP) 2015: 516–527
[ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
 
The performance of large distributed systems crucially depends on efficiently balancing their load. This has motivated a large amount of theoretical research how an imbalanced load vector can be smoothed with local algorithms. For technical reasons, the vast majority of previous work focuses on regular (or almost regular) graphs including symmetric topologies such as grids and hypercubes, and ignores the fact that large networks are often highly heterogenous. We model large scale-free networks by Chung-Lu random graphs and analyze a simple local algorithm for iterative load balancing. On n-node graphs our distributed algorithm balances the load within \(O((log~log~n)^2)\) steps. It does not need to know the exponent \(2<\beta<3\) of the power-law degree distribution or the weights \(w_i\) of the graph model. To the best of our knowledge, this is the first result which shows that load-balancing can be done in double-logarithmic time on realistic graph classes.
@inproceedings{DBLP:conf/icalp/BringmannFHRS15, abstract = {The performance of large distributed systems crucially depends on efficiently balancing their load. This has motivated a large amount of theoretical research how an imbalanced load vector can be smoothed with local algorithms. For technical reasons, the vast majority of previous work focuses on regular (or almost regular) graphs including symmetric topologies such as grids and hypercubes, and ignores the fact that large networks are often highly heterogenous. We model large scale-free networks by Chung-Lu random graphs and analyze a simple local algorithm for iterative load balancing. On n-node graphs our distributed algorithm balances the load within \(O((log&nbsp;log&nbsp;n)^2)\) steps. It does not need to know the exponent \(2<\beta<3\) of the power-law degree distribution or the weights \(w_i\) of the graph model. To the best of our knowledge, this is the first result which shows that load-balancing can be done in double-logarithmic time on realistic graph classes.}, author = {Bringmann, Karl and Friedrich, Tobias and Hoefer, Martin and Rothenberger, Ralf and Sauerwald, Thomas}, booktitle = {International Colloquium on Automata, Languages and Programming (ICALP)}, keywords = {icalp imported ralfrothenberger tobiasfriedrich year2015}, pages = {516-527}, publisher = {Springer}, series = {Lecture Notes in Computer Science}, title = {Ultra-Fast Load Balancing on Scale-Free Networks}, volume = 9135, year = 2015 }

Friedrich, Tobias; Krohmer, Anton On the Diameter of Hyperbolic Random GraphsInternational Colloquium on Automata, Languages and Programming (ICALP) 2015: 614–625
[ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
 
Large real-world networks are typically scale-free. Recent research has shown that such graphs are described best in a geometric space. More precisely, the internet can be mapped to a hyperbolic space such that geometric greedy routing performs close to optimal (Boguna, Papadopoulos, and Krioukov. Nature Communications, 1:62, 2010). This observation pushed the interest in hyperbolic networks as a natural model for scale-free networks. Hyperbolic random graphs follow a power-law degree distribution with controllable exponent \(\beta\) and show high clustering (Gugelmann, Panagiotou, and Peter. ICALP, pp. 573-585, 2012). For understanding the structure of the resulting graphs and for analyzing the behavior of network algorithms, the next question is bounding the size of the diameter. The only known explicit bound is \(O((\log n)\)\(^{32/((3-\beta)(5-\beta))})\) (Kiwi and Mitsche. ANALCO, pp. 26-39, 2015). We present two much simpler proofs for an improved upper bound of \(O((\log n)\)\(^{2/(3-\beta)})\) and a lower bound of \(\Omega(\log n)\). If the average degree is bounded from above by some constant, we show that the latter bound is tight by proving an upper bound of \(O(\log n)\).
@inproceedings{DBLP:conf/icalp/FriedrichK15, abstract = {Large real-world networks are typically scale-free. Recent research has shown that such graphs are described best in a geometric space. More precisely, the internet can be mapped to a hyperbolic space such that geometric greedy routing performs close to optimal (Boguna, Papadopoulos, and Krioukov. Nature Communications, 1:62, 2010). This observation pushed the interest in hyperbolic networks as a natural model for scale-free networks. Hyperbolic random graphs follow a power-law degree distribution with controllable exponent \(\beta\) and show high clustering (Gugelmann, Panagiotou, and Peter. ICALP, pp. 573-585, 2012). For understanding the structure of the resulting graphs and for analyzing the behavior of network algorithms, the next question is bounding the size of the diameter. The only known explicit bound is \(O((\log n)\)\(^{32/((3-\beta)(5-\beta))})\) (Kiwi and Mitsche. ANALCO, pp. 26-39, 2015). We present two much simpler proofs for an improved upper bound of \(O((\log n)\)\(^{2/(3-\beta)})\) and a lower bound of \(\Omega(\log n)\). If the average degree is bounded from above by some constant, we show that the latter bound is tight by proving an upper bound of \(O(\log n)\).}, author = {Friedrich, Tobias and Krohmer, Anton}, booktitle = {International Colloquium on Automata, Languages and Programming (ICALP)}, keywords = {antonkrohmer hyperbolic_analysis icalp imported tobiasfriedrich year2015}, pages = {614-625}, publisher = {Springer}, series = {Lecture Notes in Computer Science}, title = {On the Diameter of Hyperbolic Random Graphs}, volume = 9135, year = 2015 }

2013 [ nach oben ]

Bringmann, Karl; Friedrich, Tobias Exact and Efficient Generation of Geometric Random Variates and Random GraphsInternational Colloquium on Automata, Languages, and Programming (ICALP) 2013: 267–278
[ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
 
The standard algorithm for fast generation of ErdHo}s-Rényi random graphs only works in the Real RAM model. The critical point is the generation of geometric random variates \(Geo(p)\), for which there is no algorithm that is both exact and efficient in any bounded precision machine model. For a RAM model with word size \(w=\Omega(\log\log(1/p))\), we show that this is possible and present an exact algorithm for sampling \(Geo(p)\) in optimal expected time \(O(1 + \log(1/p) / w)\). We also give an exact algorithm for sampling \(\min\{n, Geo(p)\}\) in optimal expected time \(O(1 + \log(\min\{1/p,n\})/w)\). This yields a new exact algorithm for sampling ErdHo}s-Rényi and Chung-Lu random graphs of \(n\) vertices and \(m\) (expected) edges in optimal expected runtime \(O(n + m)\) on a RAM with word size \(w=\Theta(\log n)\).
@inproceedings{DBLP:conf/icalp/BringmannF13, abstract = {The standard algorithm for fast generation of Erd\H{o}s-Rényi random graphs only works in the Real RAM model. The critical point is the generation of geometric random variates \(Geo(p)\), for which there is no algorithm that is both exact and efficient in any bounded precision machine model. For a RAM model with word size \(w=\Omega(\log\log(1/p))\), we show that this is possible and present an exact algorithm for sampling \(Geo(p)\) in optimal expected time \(O(1 + \log(1/p) / w)\). We also give an exact algorithm for sampling \(\min\{n, Geo(p)\}\) in optimal expected time \(O(1 + \log(\min\{1/p,n\})/w)\). This yields a new exact algorithm for sampling Erd\H{o}s-Rényi and Chung-Lu random graphs of \(n\) vertices and \(m\) (expected) edges in optimal expected runtime \(O(n + m)\) on a RAM with word size \(w=\Theta(\log n)\).}, author = {Bringmann, Karl and Friedrich, Tobias}, booktitle = {International Colloquium on Automata, Languages, and Programming (ICALP)}, keywords = {icalp imported tobiasfriedrich year2013}, pages = {267-278}, publisher = {Springer}, series = {Lecture Notes in Computer Science}, title = {Exact and Efficient Generation of Geometric Random Variates and Random Graphs}, volume = 7965, year = 2013 }

2009 [ nach oben ]

Doerr, Benjamin; Friedrich, Tobias; Sauerwald, Thomas Quasirandom Rumor Spreading: Expanders, Push vs. Pull, and RobustnessInternational Colloquium on Automata, Languages and Programming (ICALP) 2009: 366–377
[ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
 
Randomized rumor spreading is an efficient protocol to distribute information in networks. Recently, a quasirandom version has been proposed and proven to work equally well on many graphs and better for sparse random graphs. In this work we show three main results for the quasirandom rumor spreading model. We exhibit a natural expansion property for networks which suffices to make quasirandom rumor spreading inform all nodes of the network in logarithmic time with high probability. This expansion property is satisfied, among others, by many expander graphs, random regular graphs, and ErdHo}s-Rényi random graphs. For all network topologies, we show that if one of the push or pull model works well, so does the other. We also show that quasirandom rumor spreading is robust against transmission failures. If each message sent out gets lost with probability \(f\), then the runtime increases only by a factor of \(O(1/(1-f))\).
@inproceedings{DBLP:conf/icalp/DoerrFS09, abstract = {Randomized rumor spreading is an efficient protocol to distribute information in networks. Recently, a quasirandom version has been proposed and proven to work equally well on many graphs and better for sparse random graphs. In this work we show three main results for the quasirandom rumor spreading model. We exhibit a natural expansion property for networks which suffices to make quasirandom rumor spreading inform all nodes of the network in logarithmic time with high probability. This expansion property is satisfied, among others, by many expander graphs, random regular graphs, and Erd\H{o}s-Rényi random graphs. For all network topologies, we show that if one of the push or pull model works well, so does the other. We also show that quasirandom rumor spreading is robust against transmission failures. If each message sent out gets lost with probability \(f\), then the runtime increases only by a factor of \(O(1/(1-f))\).}, author = {Doerr, Benjamin and Friedrich, Tobias and Sauerwald, Thomas}, booktitle = {International Colloquium on Automata, Languages and Programming (ICALP)}, keywords = {icalp imported tobiasfriedrich year2009}, pages = {366-377}, publisher = {Springer}, series = {Lecture Notes in Computer Science}, title = {Quasirandom Rumor Spreading: Expanders, Push vs. Pull, and Robustness}, volume = 5555, year = 2009 }

Friedrich, Tobias; Sauerwald, Thomas; Vilenchik, Dan Smoothed Analysis of Balancing NetworksInternational Colloquium on Automata, Languages, and Programming (ICALP) 2009: 472–483
[ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
 
In a balancing network each processor has an initial collection of unit-size jobs (tokens) and in each round, pairs of processors connected by balancers split their load as evenly as possible. An excess token (if any) is placed according to some predefined rule. As it turns out, this rule crucially affects the performance of the network. In this work we propose a model that studies this effect. We suggest a model bridging the uniformly-random assignment rule, and the arbitrary one (in the spirit of smoothed-analysis). We start with an arbitrary assignment of balancer directions and then flip each assignment with probability \(\alpha\) independently. For a large class of balancing networks our result implies that after \(O(\log n)\) rounds the discrepancy is \(O((1/2 - \alpha) \log n + \log \log n)\) with high probability. This matches and generalizes known upper bounds for \(\alpha = 0\) and \(\alpha = 1/2\). We also show that a natural network matches the upper bound for any \(\alpha\).
@inproceedings{DBLP:conf/icalp/FriedrichSV09, abstract = {In a balancing network each processor has an initial collection of unit-size jobs (tokens) and in each round, pairs of processors connected by balancers split their load as evenly as possible. An excess token (if any) is placed according to some predefined rule. As it turns out, this rule crucially affects the performance of the network. In this work we propose a model that studies this effect. We suggest a model bridging the uniformly-random assignment rule, and the arbitrary one (in the spirit of smoothed-analysis). We start with an arbitrary assignment of balancer directions and then flip each assignment with probability \(\alpha\) independently. For a large class of balancing networks our result implies that after \(O(\log n)\) rounds the discrepancy is \(O((1/2 - \alpha) \log n + \log \log n)\) with high probability. This matches and generalizes known upper bounds for \(\alpha = 0\) and \(\alpha = 1/2\). We also show that a natural network matches the upper bound for any \(\alpha\).}, author = {Friedrich, Tobias and Sauerwald, Thomas and Vilenchik, Dan}, booktitle = {International Colloquium on Automata, Languages, and Programming (ICALP)}, keywords = {icalp imported tobiasfriedrich year2009}, pages = {472-483}, publisher = {Springer}, series = {Lecture Notes in Computer Science}, title = {Smoothed Analysis of Balancing Networks}, volume = 5556, year = 2009 }