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**Published in:** LIPIcs, Volume 101, 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)

The fastest algorithms for edge coloring run in time 2^m n^{O(1)}, where m and n are the number of edges and vertices of the input graph, respectively. For dense graphs, this bound becomes 2^{Theta(n^2)}. This is a somewhat unique situation, since most of the studied graph problems admit algorithms running in time 2^{O(n log n)}. It is a notorious open problem to either show an algorithm for edge coloring running in time 2^{o(n^2)} or to refute it, assuming the Exponential Time Hypothesis (ETH) or other well established assumptions.
We notice that the same question can be asked for list edge coloring, a well-studied generalization of edge coloring where every edge comes with a set (often called a list) of allowed colors. Our main result states that list edge coloring for simple graphs does not admit an algorithm running in time 2^{o(n^2)}, unless ETH fails. Interestingly, the algorithm for edge coloring running in time 2^m n^{O(1)} generalizes to the list version without any asymptotic slow-down. Thus, our lower bound is essentially tight. This also means that in order to design an algorithm running in time 2^{o(n^2)} for edge coloring, one has to exploit its special features compared to the list version.

Lukasz Kowalik and Arkadiusz Socala. Tight Lower Bounds for List Edge Coloring. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 28:1-28:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kowalik_et_al:LIPIcs.SWAT.2018.28, author = {Kowalik, Lukasz and Socala, Arkadiusz}, title = {{Tight Lower Bounds for List Edge Coloring}}, booktitle = {16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)}, pages = {28:1--28:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-068-2}, ISSN = {1868-8969}, year = {2018}, volume = {101}, editor = {Eppstein, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2018.28}, URN = {urn:nbn:de:0030-drops-88540}, doi = {10.4230/LIPIcs.SWAT.2018.28}, annote = {Keywords: list edge coloring, complexity, ETH lower bound} }

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**Published in:** LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)

In the multicoloring problem, also known as (a:b)-coloring or b-fold coloring, we are given a graph G and a set of a colors, and the task is to assign a subset of b colors to each vertex of G so that adjacent vertices receive disjoint color subsets. This natural generalization of the classic coloring problem (the b=1 case) is equivalent to finding a homomorphism to the Kneser graph KG_{a,b}, and gives relaxations approaching the fractional chromatic number.
We study the complexity of determining whether a graph has an (a:b)-coloring. Our main result is that this problem does not admit an algorithm with running time f(b) * 2^{o(log b) n}, for any computable f(b), unless the Exponential Time Hypothesis (ETH) fails. A (b+1)^n * poly(n)-time algorithm due to Nederlof [2008] shows that this is tight. A direct corollary of our result is that the graph homomorphism problem does not admit a 2^O(n+h) algorithm unless ETH fails, even if the target graph is required to be a Kneser graph. This refines the understanding given by the recent lower bound of Cygan et al. [SODA 2016].
The crucial ingredient in our hardness reduction is the usage of detecting matrices of Lindström [Canad. Math. Bull., 1965], which is a combinatorial tool that, to the best of our knowledge, has not yet been used for proving complexity lower bounds. As a side result, we prove that the running time of the algorithms of Abasi et al. [MFCS 2014] and of Gabizon et al. [ESA 2015] for the r-monomial detection problem are optimal under ETH.

Marthe Bonamy, Lukasz Kowalik, Michal Pilipczuk, Arkadiusz Socala, and Marcin Wrochna. Tight Lower Bounds for the Complexity of Multicoloring. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bonamy_et_al:LIPIcs.ESA.2017.18, author = {Bonamy, Marthe and Kowalik, Lukasz and Pilipczuk, Michal and Socala, Arkadiusz and Wrochna, Marcin}, title = {{Tight Lower Bounds for the Complexity of Multicoloring}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {18:1--18:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-049-1}, ISSN = {1868-8969}, year = {2017}, volume = {87}, editor = {Pruhs, Kirk and Sohler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.18}, URN = {urn:nbn:de:0030-drops-78527}, doi = {10.4230/LIPIcs.ESA.2017.18}, annote = {Keywords: multicoloring, Kneser graph homomorphism, ETH lower bound} }

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**Published in:** LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)

Given a traveling salesman problem (TSP) tour H in graph G, a k-move is an operation which removes k edges from H, and adds k edges of G so that a new tour H' is formed. The popular k-opt heuristic for TSP finds a local optimum by starting from an arbitrary tour H and then improving it by a sequence of k-moves.
Until 2016, the only known algorithm to find an improving k-move for a given tour was the naive solution in time O(n^k). At ICALP'16 de Berg, Buchin, Jansen and Woeginger showed an O(n^{floor(2/3k)+1})-time algorithm.
We show an algorithm which runs in O(n^{(1/4 + epsilon_k)k}) time, where lim_{k -> infinity} epsilon_k = 0. It improves over the state of the art for every k >= 5. For the most practically relevant case k=5 we provide a slightly refined algorithm running in O(n^{3.4}) time. We also show that for the k=4 case, improving over the O(n^3)-time algorithm of de Berg et al. would be a major breakthrough: an O(n^{3 - epsilon})-time algorithm for any epsilon > 0 would imply an O(n^{3 - delta})-time algorithm for the All Pairs Shortest Paths problem, for some delta>0.

Marek Cygan, Lukasz Kowalik, and Arkadiusz Socala. Improving TSP Tours Using Dynamic Programming over Tree Decompositions. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 30:1-30:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{cygan_et_al:LIPIcs.ESA.2017.30, author = {Cygan, Marek and Kowalik, Lukasz and Socala, Arkadiusz}, title = {{Improving TSP Tours Using Dynamic Programming over Tree Decompositions}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {30:1--30:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-049-1}, ISSN = {1868-8969}, year = {2017}, volume = {87}, editor = {Pruhs, Kirk and Sohler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.30}, URN = {urn:nbn:de:0030-drops-78539}, doi = {10.4230/LIPIcs.ESA.2017.30}, annote = {Keywords: TSP, treewidth, local search, XP algorithm, hardness in P} }

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**Published in:** LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

The Rainbow k-Coloring problem asks whether the edges of a given graph can be colored in k colors so that every pair of vertices is connected by a rainbow path, i.e., a path with all edges of different colors. Our main result states that for any k >= 2, there is no algorithm for Rainbow k-Coloring running in time 2^{o(n^{3/2})}, unless ETH fails. Motivated by this negative result we consider two parameterized variants of the problem. In the Subset Rainbow k-Coloring problem, introduced by Chakraborty et al. [STACS 2009, J. Comb. Opt. 2009], we are additionally given a set S of pairs of vertices and we ask if there is a coloring in which all the pairs in S are connected by rainbow paths. We show that Subset Rainbow k-Coloring is FPT when parameterized by |S|. We also study Subset Rainbow k-Coloring problem, where we are additionally given an integer q and we ask if there is a coloring in which at least q anti-edges are connected by rainbow paths. We show that the problem is FPT when parameterized by q and has a kernel of size O(q) for every k >= 2, extending the result of Ananth et al. [FSTTCS 2011]. We believe that our techniques used for the lower bounds may shed some light on the complexity of the classical Edge Coloring problem, where it is a major open question if a 2^{O(n)}-time algorithm exists.

Lukasz Kowalik, Juho Lauri, and Arkadiusz Socala. On the Fine-Grained Complexity of Rainbow Coloring. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 58:1-58:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kowalik_et_al:LIPIcs.ESA.2016.58, author = {Kowalik, Lukasz and Lauri, Juho and Socala, Arkadiusz}, title = {{On the Fine-Grained Complexity of Rainbow Coloring}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {58:1--58:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.58}, URN = {urn:nbn:de:0030-drops-64001}, doi = {10.4230/LIPIcs.ESA.2016.58}, annote = {Keywords: graph coloring, computational complexity, lower bounds, exponential time hypothesis, FPT algorithms} }

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**Published in:** LIPIcs, Volume 43, 10th International Symposium on Parameterized and Exact Computation (IPEC 2015)

In the k-Leaf Out-Branching and k-Internal Out-Branching problems we are given a directed graph D with a designated root r and a nonnegative integer k. The question is to determine the existence of an outbranching rooted at r that has at least k leaves, or at least k internal vertices, respectively. Both these problems were intensively studied from the points of view of parameterized complexity and kernelization, and in particular for both of them kernels with O(k^2) vertices are known on general graphs. In this work we show that k-Leaf Out-Branching admits a kernel with O(k) vertices on H-minor-free graphs, for any fixed H, whereas k-Internal Out-Branching admits a kernel with O(k) vertices on any graph class of bounded expansion.

Marthe Bonamy, Lukasz Kowalik, Michal Pilipczuk, and Arkadiusz Socala. Linear Kernels for Outbranching Problems in Sparse Digraphs. In 10th International Symposium on Parameterized and Exact Computation (IPEC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 43, pp. 199-211, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bonamy_et_al:LIPIcs.IPEC.2015.199, author = {Bonamy, Marthe and Kowalik, Lukasz and Pilipczuk, Michal and Socala, Arkadiusz}, title = {{Linear Kernels for Outbranching Problems in Sparse Digraphs}}, booktitle = {10th International Symposium on Parameterized and Exact Computation (IPEC 2015)}, pages = {199--211}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-92-7}, ISSN = {1868-8969}, year = {2015}, volume = {43}, editor = {Husfeldt, Thore and Kanj, Iyad}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2015.199}, URN = {urn:nbn:de:0030-drops-55839}, doi = {10.4230/LIPIcs.IPEC.2015.199}, annote = {Keywords: FPT algorithm, kernelization, outbranching, sparse graphs} }

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**Published in:** LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)

We show an O^*(2^k)-time polynomial space algorithm for the k-sized Graph Motif problem. We also introduce a new optimization variant of the problem, called Closest Graph Motif and solve it within the same time bound. The Closest Graph Motif problem encompasses several previously studied optimization variants, like Maximum Graph Motif, Min-Substitute, and Min-Add.
Moreover, we provide a piece of evidence that our result might be essentially tight: the existence of an O^*((2-epsilon)^k)-time algorithm for the Graph Motif problem implies an ((2-epsilon')^n)-time algorithm for Set Cover.

Andreas Björklund, Petteri Kaski, and Lukasz Kowalik. Probably Optimal Graph Motifs. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 20-31, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{bjorklund_et_al:LIPIcs.STACS.2013.20, author = {Bj\"{o}rklund, Andreas and Kaski, Petteri and Kowalik, Lukasz}, title = {{Probably Optimal Graph Motifs}}, booktitle = {30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)}, pages = {20--31}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-50-7}, ISSN = {1868-8969}, year = {2013}, volume = {20}, editor = {Portier, Natacha and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.20}, URN = {urn:nbn:de:0030-drops-39196}, doi = {10.4230/LIPIcs.STACS.2013.20}, annote = {Keywords: graph motif, FPT algorithm} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 8431, Moderately Exponential Time Algorithms (2008)

Two problem sessions were part of the seminar on Moderately Exponential Time Algorithms. Some of the open problems presented at those sessions have been collected.

Fedor V. Fomin, Kazuo Iwama, Dieter Kratsch, Petteri Kaski, Mikko Koivisto, Lukasz Kowalik, Yoshio Okamoto, Johan van Rooij, and Ryan Williams. 08431 Open Problems – Moderately Exponential Time Algorithms. In Moderately Exponential Time Algorithms. Dagstuhl Seminar Proceedings, Volume 8431, pp. 1-8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{fomin_et_al:DagSemProc.08431.3, author = {Fomin, Fedor V. and Iwama, Kazuo and Kratsch, Dieter and Kaski, Petteri and Koivisto, Mikko and Kowalik, Lukasz and Okamoto, Yoshio and van Rooij, Johan and Williams, Ryan}, title = {{08431 Open Problems – Moderately Exponential Time Algorithms}}, booktitle = {Moderately Exponential Time Algorithms}, pages = {1--8}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2008}, volume = {8431}, editor = {Fedor V. Fomin and Kazuo Iwama and Dieter Kratsch}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.08431.3}, URN = {urn:nbn:de:0030-drops-17986}, doi = {10.4230/DagSemProc.08431.3}, annote = {Keywords: Algorithms, NP-hard problems, Moderately Exponential Time Algorithms} }

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