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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 Number. International Colloquium on Automata, Languages and Programming (ICALP) 2019
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 graphparameters, cutwidth and pathwidth. These connections allow us to show that computing the locality number is NPhard 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 byproduct, 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 greedybased approximation algorithms for the locality number.

Friedrich, Tobias; Rothenberger, Ralf The Satisfiability Threshold for NonUniform Random 2SAT. International Colloquium on Automata, Languages and Programming (ICALP) 2019
Propositional satisfiability (SAT) is one of the most fundamental problems in computer science. Its worstcase hardness lies at the core of computational complexity theory, for example in the form of NPhardness and the (Strong) Exponential Time Hypothesis. In practice however, SAT instances can often be solved efficiently. This contradicting behavior has spawned interest in the averagecase analysis of SAT and has triggered the development of sophisticated rigorous and nonrigorous 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, realworld instances often exhibit large fluctuations in variable occurrence. This can be modeled by a nonuniform distribution of the variables, which can result in distributions closer to industrial SAT instances. We study satisfiability thresholds of nonuniform random 2SAT with n variables and m clauses and with an arbitrary probability distribution \((p_i)_{i \in [n]}\) with \(p_1 \geq p_2 \geq \dots \geq p_n > 0\) 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 \(m = \Theta((1− \sum_{i=1^n p_i^2) / (p_1 cdot (\sum_{i=2^n p_i^2)^{1/2}))\) 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].

Casel, Katrin; Fernau, Henning; Gaspers, Serge; Gras, Benjamin; Schmid, Markus L. On the Complexity of GrammarBased Compression over Fixed Alphabets. International Colloquium on Automata, Languages, and Programming (ICALP) 2016: 122:1122:14
It is shown that the shortestgrammar problem remains NPcomplete 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 polynomialtime, 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(3n) exact exponentialtime algorithm, based on dynamic programming. Similar results are also given for 1level 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 shortestgrammar problem).

Galanis, Andreas; Göbel, Andreas; Goldberg, LeslieAnn; Lapinskas, John; Richerby, David Amplifiers for the Moran Process. International Colloquium on Automata, Languages and Programming (ICALP) 2016: 62:162:13
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 nonmutant) is passed on to an outneighbour 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 uniformlyplaced 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 stronglyamplifying 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 nvertex graph in this family, starting with a uniformlychosen 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. Theoremnumbering here matches the full version.

Göbel, Andreas; Goldberg, Leslie Ann; Richerby, David Counting Homomorphisms to SquareFree Graphs, Modulo 2. International Colloquium on Automata, Languages, and Programming (ICALP) 2015: 642653
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 4cycles, \(\oplus\)HomsToH is either in polynomial time or is \(\oplus\)Pcomplete. This partially confirms a conjecture of Faben and Jerrum that was previously only known to hold for trees and for a restricted class of treewidth2 graphs called cactus graphs. We confirm the conjecture for a rich class of graphs, including graphs of unbounded treewidth. In particular, we focus on squarefree graphs, which are graphs without 4cycles. 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 treelike so that treelike 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.

Friedrich, Tobias; Krohmer, Anton On the Diameter of Hyperbolic Random Graphs. International Colloquium on Automata, Languages and Programming (ICALP) 2015: 614625
Large realworld networks are typically scalefree. 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 scalefree networks. Hyperbolic random graphs follow a powerlaw degree distribution with controllable exponent \(\beta\) and show high clustering (Gugelmann, Panagiotou, and Peter. ICALP, pp. 573585, 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. 2639, 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 \(O(\log n)\).

Bringmann, Karl; Friedrich, Tobias; Hoefer, Martin; Rothenberger, Ralf; Sauerwald, Thomas UltraFast Load Balancing on ScaleFree Networks. International Colloquium on Automata, Languages and Programming (ICALP) 2015: 516527
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 scalefree networks by ChungLu random graphs and analyze a simple local algorithm for iterative load balancing. On nnode graphs our distributed algorithm balances the load within \(O((\log~\log~n)^2)\) steps. It does not need to know the exponent \(beta in (2,3)\) of the powerlaw 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 loadbalancing can be done in doublelogarithmic time on realistic graph classes.

Doerr, Benjamin; Friedrich, Tobias; Sauerwald, Thomas Quasirandom Rumor Spreading: Expanders, Push vs. Pull, and Robustness. International Colloquium on Automata, Languages and Programming (ICALP) 2009: 366377
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}sRényi 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 \(O(1/(1f))\).

Friedrich, Tobias; Sauerwald, Thomas; Vilenchik, Dan Smoothed Analysis of Balancing Networks. International Colloquium on Automata, Languages, and Programming (ICALP) 2009: 472483
In a balancing network each processor has an initial collection of unitsize 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 uniformlyrandom assignment rule, and the arbitrary one (in the spirit of smoothedanalysis). 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 \(\alpha\).