12 Search Results for "Kowalik, Lukasz"


Document
An Asymptotically Fast Polynomial Space Algorithm for Hamiltonicity Detection in Sparse Directed Graphs

Authors: Andreas Björklund

Published in: LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)


Abstract
We present a polynomial space Monte Carlo algorithm that given a directed graph on n vertices and average outdegree δ, detects if the graph has a Hamiltonian cycle in 2^{n-Ω(n/δ)} time. This asymptotic scaling of the savings in the running time matches the fastest known exponential space algorithm by Björklund and Williams ICALP 2019. By comparison, the previously best polynomial space algorithm by Kowalik and Majewski IPEC 2020 guarantees a 2^{n-Ω(n/(2^δ))} time bound. Our algorithm combines for the first time the idea of obtaining a fingerprint of the presence of a Hamiltonian cycle through an inclusion-exclusion summation over the Laplacian of the graph from Björklund, Kaski, and Koutis ICALP 2017, with the idea of sieving for the non-zero terms in an inclusion-exclusion summation by listing solutions to systems of linear equations over ℤ₂ from Björklund and Husfeldt FOCS 2013.

Cite as

Andreas Björklund. An Asymptotically Fast Polynomial Space Algorithm for Hamiltonicity Detection in Sparse Directed Graphs. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 15:1-15:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{bjorklund:LIPIcs.STACS.2021.15,
  author =	{Bj\"{o}rklund, Andreas},
  title =	{{An Asymptotically Fast Polynomial Space Algorithm for Hamiltonicity Detection in Sparse Directed Graphs}},
  booktitle =	{38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
  pages =	{15:1--15:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-180-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{187},
  editor =	{Bl\"{a}ser, Markus and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.15},
  URN =		{urn:nbn:de:0030-drops-136601},
  doi =		{10.4230/LIPIcs.STACS.2021.15},
  annote =	{Keywords: Hamiltonian cycle, directed graph, worst case analysis algorithm}
}
Document
The Asymmetric Travelling Salesman Problem In Sparse Digraphs

Authors: Łukasz Kowalik and Konrad Majewski

Published in: LIPIcs, Volume 180, 15th International Symposium on Parameterized and Exact Computation (IPEC 2020)


Abstract
Asymmetric Travelling Salesman Problem (ATSP) and its special case Directed Hamiltonicity are among the most fundamental problems in computer science. The dynamic programming algorithm running in time 𝒪^*(2ⁿ) developed almost 60 years ago by Bellman, Held and Karp, is still the state of the art for both of these problems. In this work we focus on sparse digraphs. First, we recall known approaches for Undirected Hamiltonicity and TSP in sparse graphs and we analyse their consequences for Directed Hamiltonicity and ATSP in sparse digraphs, either by adapting the algorithm, or by using reductions. In this way, we get a number of running time upper bounds for a few classes of sparse digraphs, including 𝒪^*(2^(n/3)) for digraphs with both out- and indegree bounded by 2, and 𝒪^*(3^(n/2)) for digraphs with outdegree bounded by 3. Our main results are focused on digraphs of bounded average outdegree d. The baseline for ATSP here is a simple enumeration of cycle covers which can be done in time bounded by 𝒪^*(μ(d)ⁿ) for a function μ(d) ≤ (⌈d⌉!)^(1/⌈d⌉). One can also observe that Directed Hamiltonicity can be solved in randomized time 𝒪^*((2-2^(-d))ⁿ) and polynomial space, by adapting a recent result of Björklund [ISAAC 2018] stated originally for Undirected Hamiltonicity in sparse bipartite graphs. We present two new deterministic algorithms for ATSP: the first running in time 𝒪(2^(0.441(d-1)n)) and polynomial space, and the second in exponential space with running time of 𝒪^*(τ(d)^(n/2)) for a function τ(d) ≤ d.

Cite as

Łukasz Kowalik and Konrad Majewski. The Asymmetric Travelling Salesman Problem In Sparse Digraphs. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 23:1-23:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{kowalik_et_al:LIPIcs.IPEC.2020.23,
  author =	{Kowalik, {\L}ukasz and Majewski, Konrad},
  title =	{{The Asymmetric Travelling Salesman Problem In Sparse Digraphs}},
  booktitle =	{15th International Symposium on Parameterized and Exact Computation (IPEC 2020)},
  pages =	{23:1--23:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-172-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{180},
  editor =	{Cao, Yixin and Pilipczuk, Marcin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2020.23},
  URN =		{urn:nbn:de:0030-drops-133269},
  doi =		{10.4230/LIPIcs.IPEC.2020.23},
  annote =	{Keywords: asymmetric traveling salesman problem, Hamiltonian cycle, sparse graphs, exponential algorithm}
}
Document
The PACE 2020 Parameterized Algorithms and Computational Experiments Challenge: Treedepth

Authors: Łukasz Kowalik, Marcin Mucha, Wojciech Nadara, Marcin Pilipczuk, Manuel Sorge, and Piotr Wygocki

Published in: LIPIcs, Volume 180, 15th International Symposium on Parameterized and Exact Computation (IPEC 2020)


Abstract
This year’s Parameterized Algorithms and Computational Experiments challenge (PACE 2020) was devoted to the problem of computing the treedepth of a given graph. Altogether 51 participants from 20 teams, 12 countries and 3 continents submitted their implementations to the competition. In this report, we describe the setup of the challenge, the selection of benchmark instances and the ranking of the participating teams. We also briefly discuss the approaches used in the submitted solvers and the differences in their performance on our benchmark dataset.

Cite as

Łukasz Kowalik, Marcin Mucha, Wojciech Nadara, Marcin Pilipczuk, Manuel Sorge, and Piotr Wygocki. The PACE 2020 Parameterized Algorithms and Computational Experiments Challenge: Treedepth. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 37:1-37:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{kowalik_et_al:LIPIcs.IPEC.2020.37,
  author =	{Kowalik, {\L}ukasz and Mucha, Marcin and Nadara, Wojciech and Pilipczuk, Marcin and Sorge, Manuel and Wygocki, Piotr},
  title =	{{The PACE 2020 Parameterized Algorithms and Computational Experiments Challenge: Treedepth}},
  booktitle =	{15th International Symposium on Parameterized and Exact Computation (IPEC 2020)},
  pages =	{37:1--37:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-172-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{180},
  editor =	{Cao, Yixin and Pilipczuk, Marcin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2020.37},
  URN =		{urn:nbn:de:0030-drops-133404},
  doi =		{10.4230/LIPIcs.IPEC.2020.37},
  annote =	{Keywords: computing treedepth, contest, implementation challenge, FPT}
}
Document
Many Visits TSP Revisited

Authors: Łukasz Kowalik, Shaohua Li, Wojciech Nadara, Marcin Smulewicz, and Magnus Wahlström

Published in: LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)


Abstract
We study the Many Visits TSP problem, where given a number k(v) for each of n cities and pairwise (possibly asymmetric) integer distances, one has to find an optimal tour that visits each city v exactly k(v) times. The currently fastest algorithm is due to Berger, Kozma, Mnich and Vincze [SODA 2019, TALG 2020] and runs in time and space O*(5ⁿ). They also show a polynomial space algorithm running in time O(16^{n+o(n)}). In this work, we show three main results: - A randomized polynomial space algorithm in time O*(2^n D), where D is the maximum distance between two cities. By using standard methods, this results in a (1+ε)-approximation in time O*(2ⁿε^{-1}). Improving the constant 2 in these results would be a major breakthrough, as it would result in improving the O*(2ⁿ)-time algorithm for Directed Hamiltonian Cycle, which is a 50 years old open problem. - A tight analysis of Berger et al.’s exponential space algorithm, resulting in an O*(4ⁿ) running time bound. - A new polynomial space algorithm, running in time O(7.88ⁿ).

Cite as

Łukasz Kowalik, Shaohua Li, Wojciech Nadara, Marcin Smulewicz, and Magnus Wahlström. Many Visits TSP Revisited. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 66:1-66:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{kowalik_et_al:LIPIcs.ESA.2020.66,
  author =	{Kowalik, {\L}ukasz and Li, Shaohua and Nadara, Wojciech and Smulewicz, Marcin and Wahlstr\"{o}m, Magnus},
  title =	{{Many Visits TSP Revisited}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{66:1--66:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.66},
  URN =		{urn:nbn:de:0030-drops-129329},
  doi =		{10.4230/LIPIcs.ESA.2020.66},
  annote =	{Keywords: many visits traveling salesman problem, exponential algorithm}
}
Document
Fine-Grained Complexity of k-OPT in Bounded-Degree Graphs for Solving TSP

Authors: Édouard Bonnet, Yoichi Iwata, Bart M. P. Jansen, and Łukasz Kowalik

Published in: LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)


Abstract
The Traveling Salesman Problem asks to find a minimum-weight Hamiltonian cycle in an edge-weighted complete graph. Local search is a widely-employed strategy for finding good solutions to TSP. A popular neighborhood operator for local search is k-opt, which turns a Hamiltonian cycle C into a new Hamiltonian cycle C' by replacing k edges. We analyze the problem of determining whether the weight of a given cycle can be decreased by a k-opt move. Earlier work has shown that (i) assuming the Exponential Time Hypothesis, there is no algorithm that can detect whether or not a given Hamiltonian cycle C in an n-vertex input can be improved by a k-opt move in time f(k) n^o(k / log k) for any function f, while (ii) it is possible to improve on the brute-force running time of O(n^k) and save linear factors in the exponent. Modern TSP heuristics are very successful at identifying the most promising edges to be used in k-opt moves, and experiments show that very good global solutions can already be reached using only the top-O(1) most promising edges incident to each vertex. This leads to the following question: can improving k-opt moves be found efficiently in graphs of bounded degree? We answer this question in various regimes, presenting new algorithms and conditional lower bounds. We show that the aforementioned ETH lower bound also holds for graphs of maximum degree three, but that in bounded-degree graphs the best improving k-move can be found in time O(n^((23/135+epsilon_k)k)), where lim_{k -> infty} epsilon_k = 0. This improves upon the best-known bounds for general graphs. Due to its practical importance, we devote special attention to the range of k in which improving k-moves in bounded-degree graphs can be found in quasi-linear time. For k <= 7, we give quasi-linear time algorithms for general weights. For k=8 we obtain a quasi-linear time algorithm when the weights are bounded by O(polylog n). On the other hand, based on established fine-grained complexity hypotheses about the impossibility of detecting a triangle in edge-linear time, we prove that the k = 9 case does not admit quasi-linear time algorithms. Hence we fully characterize the values of k for which quasi-linear time algorithms exist for polylogarithmic weights on bounded-degree graphs.

Cite as

Édouard Bonnet, Yoichi Iwata, Bart M. P. Jansen, and Łukasz Kowalik. Fine-Grained Complexity of k-OPT in Bounded-Degree Graphs for Solving TSP. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 23:1-23:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{bonnet_et_al:LIPIcs.ESA.2019.23,
  author =	{Bonnet, \'{E}douard and Iwata, Yoichi and Jansen, Bart M. P. and Kowalik, {\L}ukasz},
  title =	{{Fine-Grained Complexity of k-OPT in Bounded-Degree Graphs for Solving TSP}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{23:1--23:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Bender, Michael A. and Svensson, Ola and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.23},
  URN =		{urn:nbn:de:0030-drops-111444},
  doi =		{10.4230/LIPIcs.ESA.2019.23},
  annote =	{Keywords: traveling salesman problem, k-OPT, bounded degree}
}
Document
Tight Lower Bounds for List Edge Coloring

Authors: Lukasz Kowalik and Arkadiusz Socala

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


Abstract
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.

Cite as

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}
}
Document
Tight Lower Bounds for the Complexity of Multicoloring

Authors: Marthe Bonamy, Lukasz Kowalik, Michal Pilipczuk, Arkadiusz Socala, and Marcin Wrochna

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


Abstract
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.

Cite as

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}
}
Document
Improving TSP Tours Using Dynamic Programming over Tree Decompositions

Authors: Marek Cygan, Lukasz Kowalik, and Arkadiusz Socala

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


Abstract
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.

Cite as

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}
}
Document
On the Fine-Grained Complexity of Rainbow Coloring

Authors: Lukasz Kowalik, Juho Lauri, and Arkadiusz Socala

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


Abstract
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.

Cite as

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}
}
Document
Linear Kernels for Outbranching Problems in Sparse Digraphs

Authors: Marthe Bonamy, Lukasz Kowalik, Michal Pilipczuk, and Arkadiusz Socala

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


Abstract
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.

Cite as

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}
}
Document
Probably Optimal Graph Motifs

Authors: Andreas Björklund, Petteri Kaski, and Lukasz Kowalik

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


Abstract
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.

Cite as

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}
}
Document
08431 Open Problems – Moderately Exponential Time Algorithms

Authors: Fedor V. Fomin, Kazuo Iwama, Dieter Kratsch, Petteri Kaski, Mikko Koivisto, Lukasz Kowalik, Yoshio Okamoto, Johan van Rooij, and Ryan Williams

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


Abstract
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.

Cite as

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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