9 Search Results for "Belmonte, Rémy"

Document
DOI: 10.4230/LIPIcs.WABI.2024.16
Finding Maximum Common Contractions Between Phylogenetic Networks

Authors: Bertrand Marchand, Nadia Tahiri, Olivier Tremblay-Savard, and Manuel Lafond

Published in: LIPIcs, Volume 312, 24th International Workshop on Algorithms in Bioinformatics (WABI 2024)

Abstract
In this paper, we lay the groundwork on the comparison of phylogenetic networks based on edge contractions and expansions as edit operations, as originally proposed by Robinson and Foulds to compare trees. We prove that these operations connect the space of all phylogenetic networks on the same set of leaves, even if we forbid contractions that create cycles. This allows to define an operational distance on this space, as the minimum number of contractions and expansions required to transform one network into another. We highlight the difference between this distance and the computation of the maximum common contraction between two networks. Given its ability to outline a common structure between them, which can provide valuable biological insights, we study the algorithmic aspects of the latter. We first prove that computing a maximum common contraction between two networks is NP-hard, even when the maximum degree, the size of the common contraction, or the number of leaves is bounded. We also provide lower bounds to the problem based on the Exponential-Time Hypothesis. Nonetheless, we do provide a polynomial-time algorithm for weakly galled trees, a generalization of galled trees.
Cite as

Bertrand Marchand, Nadia Tahiri, Olivier Tremblay-Savard, and Manuel Lafond. Finding Maximum Common Contractions Between Phylogenetic Networks. In 24th International Workshop on Algorithms in Bioinformatics (WABI 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 312, pp. 16:1-16:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{marchand_et_al:LIPIcs.WABI.2024.16,
  author =	{Marchand, Bertrand and Tahiri, Nadia and Tremblay-Savard, Olivier and Lafond, Manuel},
  title =	{{Finding Maximum Common Contractions Between Phylogenetic Networks}},
  booktitle =	{24th International Workshop on Algorithms in Bioinformatics (WABI 2024)},
  pages =	{16:1--16:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-340-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{312},
  editor =	{Pissis, Solon P. and Sung, Wing-Kin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2024.16},
  URN =		{urn:nbn:de:0030-drops-206606},
  doi =		{10.4230/LIPIcs.WABI.2024.16},
  annote =	{Keywords: Phylogenetic networks, contractions, algorithms, weakly galled trees}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2024.16
Tractability of Packing Vertex-Disjoint A-Paths Under Length Constraints

Authors: Susobhan Bandopadhyay, Aritra Banik, Diptapriyo Majumdar, and Abhishek Sahu

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
Given an undirected graph G and a set A ⊆ V(G), an A-path is a path in G that starts and ends at two distinct vertices of A with intermediate vertices in V(G)⧵A. An A-path is called an (A,𝓁)-path if the length of the path is exactly 𝓁. In the (A, 𝓁)-Path Packing problem (ALPP), we seek to determine whether there exist k vertex-disjoint (A, 𝓁)-paths in G or not. The problem is already known to be fixed-parmeter tractable when parameterized by k+𝓁 via color coding while it remains Para-NP-hard when parameterized by k (Hamiltonian Path) or 𝓁 (P₃-Partition) alone. Therefore, a logical direction to pursue this problem is to examine it in relation to structural parameters. Belmonte et al. initiated a study along these lines and proved that ALPP parameterized by pw+|A| is W[1]-hard where pw is the pathwidth of G. In this paper, we strengthen their result and prove that it is unlikely that ALPP is fixed-parameter tractable even with respect to a bigger parameter (|A|+dtp) where dtp denotes the distance between G and a path graph (distance to path). We use a randomized reduction to achieve the mentioned result. Toward this, we prove a lemma similar to the influential "isolation lemma": Given a set system (X,ℱ) if the elements of X are assigned a weight uniformly at random from a set of values fairly large, then each subset in ℱ will have a unique weight with high probability. We believe that this result will be useful beyond the scope of this paper. ALPP being hard even for structural parameters like distance to path+|A| rules out the possibility of any FPT algorithms for many well-known other structural parameters, including FVS+|A| and treewidth+|A|. There is a straightforward FPT algorithm for ALPP parameterized by vc, the vertex cover number of the input graph. Following this, we consider the parameters CVD(cluster vertex deletion)+|A| and CVD+|𝓁| and show the problem to be FPT with respect to these parameters. Note that CVD is incomparable to the treewidth of a graph and has been in vogue recently.
Cite as

Susobhan Bandopadhyay, Aritra Banik, Diptapriyo Majumdar, and Abhishek Sahu. Tractability of Packing Vertex-Disjoint A-Paths Under Length Constraints. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 16:1-16:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bandopadhyay_et_al:LIPIcs.MFCS.2024.16,
  author =	{Bandopadhyay, Susobhan and Banik, Aritra and Majumdar, Diptapriyo and Sahu, Abhishek},
  title =	{{Tractability of Packing Vertex-Disjoint A-Paths Under Length Constraints}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{16:1--16:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.16},
  URN =		{urn:nbn:de:0030-drops-205725},
  doi =		{10.4230/LIPIcs.MFCS.2024.16},
  annote =	{Keywords: Parameterized complexity, (A,𝓁)-Path Packing, Kernelization, Randomized-Exponential Time Hypothesis, Graph Classes}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2024.58
Parameterized Vertex Integrity Revisited

Authors: Tesshu Hanaka, Michael Lampis, Manolis Vasilakis, and Kanae Yoshiwatari

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
Vertex integrity is a graph parameter that measures the connectivity of a graph. Informally, its meaning is that a graph has small vertex integrity if it has a small separator whose removal disconnects the graph into connected components which are themselves also small. Graphs with low vertex integrity are very structured; this renders many hard problems tractable and has recently attracted interest in this notion from the parameterized complexity community. In this paper we revisit the NP-complete problem of computing the vertex integrity of a given graph from the point of view of structural parameterizations. We present a number of new results, which also answer some recently posed open questions from the literature. Specifically, we show that unweighted vertex integrity is W[1]-hard parameterized by treedepth; we show that the problem remains W[1]-hard if we parameterize by feedback edge set size (via a reduction from a Bin Packing variant which may be of independent interest); and complementing this we show that the problem is FPT by max-leaf number. Furthermore, for weighted vertex integrity, we show that the problem admits a single-exponential FPT algorithm parameterized by vertex cover or by modular width, the latter result improving upon a previous algorithm which required weights to be polynomially bounded.
Cite as

Tesshu Hanaka, Michael Lampis, Manolis Vasilakis, and Kanae Yoshiwatari. Parameterized Vertex Integrity Revisited. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 58:1-58:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{hanaka_et_al:LIPIcs.MFCS.2024.58,
  author =	{Hanaka, Tesshu and Lampis, Michael and Vasilakis, Manolis and Yoshiwatari, Kanae},
  title =	{{Parameterized Vertex Integrity Revisited}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{58:1--58:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.58},
  URN =		{urn:nbn:de:0030-drops-206141},
  doi =		{10.4230/LIPIcs.MFCS.2024.58},
  annote =	{Keywords: Parameterized Complexity, Treedepth, Vertex Integrity}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.64
Hedonic Games and Treewidth Revisited

Authors: Tesshu Hanaka and Michael Lampis

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
We revisit the complexity of the well-studied notion of Additively Separable Hedonic Games (ASHGs). Such games model a basic clustering or coalition formation scenario in which selfish agents are represented by the vertices of an edge-weighted digraph G = (V,E), and the weight of an arc uv denotes the utility u gains by being in the same coalition as v. We focus on (arguably) the most basic stability question about such a game: given a graph, does a Nash stable solution exist and can we find it efficiently? We study the (parameterized) complexity of ASHG stability when the underlying graph has treewidth t and maximum degree Δ. The current best FPT algorithm for this case was claimed by Peters [AAAI 2016], with time complexity roughly 2^{O(Δ⁵t)}. We present an algorithm with parameter dependence (Δ t)^{O(Δ t)}, significantly improving upon the parameter dependence on Δ given by Peters, albeit with a slightly worse dependence on t. Our main result is that this slight performance deterioration with respect to t is actually completely justified: we observe that the previously claimed algorithm is incorrect, and that in fact no algorithm can achieve dependence t^{o(t)} for bounded-degree graphs, unless the ETH fails. This, together with corresponding bounds we provide on the dependence on Δ and the joint parameter establishes that our algorithm is essentially optimal for both parameters, under the ETH. We then revisit the parameterization by treewidth alone and resolve a question also posed by Peters by showing that Nash Stability remains strongly NP-hard on stars under additive preferences. Nevertheless, we also discover an island of mild tractability: we show that Connected Nash Stability is solvable in pseudo-polynomial time for constant t, though with an XP dependence on t which, as we establish, cannot be avoided.
Cite as

Tesshu Hanaka and Michael Lampis. Hedonic Games and Treewidth Revisited. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 64:1-64:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{hanaka_et_al:LIPIcs.ESA.2022.64,
  author =	{Hanaka, Tesshu and Lampis, Michael},
  title =	{{Hedonic Games and Treewidth Revisited}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{64:1--64:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.64},
  URN =		{urn:nbn:de:0030-drops-170025},
  doi =		{10.4230/LIPIcs.ESA.2022.64},
  annote =	{Keywords: Hedonic Games, Nash Equilibrium, Treewidth}
}
Document
DOI: 10.4230/LIPIcs.ESA.2020.14
Grundy Distinguishes Treewidth from Pathwidth

Authors: Rémy Belmonte, Eun Jung Kim, Michael Lampis, Valia Mitsou, and Yota Otachi

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

Abstract
Structural graph parameters, such as treewidth, pathwidth, and clique-width, are a central topic of study in parameterized complexity. A main aim of research in this area is to understand the "price of generality" of these widths: as we transition from more restrictive to more general notions, which are the problems that see their complexity status deteriorate from fixed-parameter tractable to intractable? This type of question is by now very well-studied, but, somewhat strikingly, the algorithmic frontier between the two (arguably) most central width notions, treewidth and pathwidth, is still not understood: currently, no natural graph problem is known to be W-hard for one but FPT for the other. Indeed, a surprising development of the last few years has been the observation that for many of the most paradigmatic problems, their complexities for the two parameters actually coincide exactly, despite the fact that treewidth is a much more general parameter. It would thus appear that the extra generality of treewidth over pathwidth often comes "for free". Our main contribution in this paper is to uncover the first natural example where this generality comes with a high price. We consider Grundy Coloring, a variation of coloring where one seeks to calculate the worst possible coloring that could be assigned to a graph by a greedy First-Fit algorithm. We show that this well-studied problem is FPT parameterized by pathwidth; however, it becomes significantly harder (W[1]-hard) when parameterized by treewidth. Furthermore, we show that Grundy Coloring makes a second complexity jump for more general widths, as it becomes para-NP-hard for clique-width. Hence, Grundy Coloring nicely captures the complexity trade-offs between the three most well-studied parameters. Completing the picture, we show that Grundy Coloring is FPT parameterized by modular-width.
Cite as

Rémy Belmonte, Eun Jung Kim, Michael Lampis, Valia Mitsou, and Yota Otachi. Grundy Distinguishes Treewidth from Pathwidth. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 14:1-14:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{belmonte_et_al:LIPIcs.ESA.2020.14,
  author =	{Belmonte, R\'{e}my and Kim, Eun Jung and Lampis, Michael and Mitsou, Valia and Otachi, Yota},
  title =	{{Grundy Distinguishes Treewidth from Pathwidth}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{14:1--14:19},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.14},
  URN =		{urn:nbn:de:0030-drops-128803},
  doi =		{10.4230/LIPIcs.ESA.2020.14},
  annote =	{Keywords: Treewidth, Pathwidth, Clique-width, Grundy Coloring}
}
Document
DOI: 10.4230/LIPIcs.STACS.2019.13
Token Sliding on Split Graphs

Authors: Rémy Belmonte, Eun Jung Kim, Michael Lampis, Valia Mitsou, Yota Otachi, and Florian Sikora

Published in: LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

Abstract
We consider the complexity of the Independent Set Reconfiguration problem under the Token Sliding rule. In this problem we are given two independent sets of a graph and are asked if we can transform one to the other by repeatedly exchanging a vertex that is currently in the set with one of its neighbors, while maintaining the set independent. Our main result is to show that this problem is PSPACE-complete on split graphs (and hence also on chordal graphs), thus resolving an open problem in this area. We then go on to consider the c-Colorable Reconfiguration problem under the same rule, where the constraint is now to maintain the set c-colorable at all times. As one may expect, a simple modification of our reduction shows that this more general problem is PSPACE-complete for all fixed c >= 1 on chordal graphs. Somewhat surprisingly, we show that the same cannot be said for split graphs: we give a polynomial time (n^{O(c)}) algorithm for all fixed values of c, except c=1, for which the problem is PSPACE-complete. We complement our algorithm with a lower bound showing that c-Colorable Reconfiguration is W[2]-hard on split graphs parameterized by c and the length of the solution, as well as a tight ETH-based lower bound for both parameters.
Cite as

Rémy Belmonte, Eun Jung Kim, Michael Lampis, Valia Mitsou, Yota Otachi, and Florian Sikora. Token Sliding on Split Graphs. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 13:1-13:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{belmonte_et_al:LIPIcs.STACS.2019.13,
  author =	{Belmonte, R\'{e}my and Kim, Eun Jung and Lampis, Michael and Mitsou, Valia and Otachi, Yota and Sikora, Florian},
  title =	{{Token Sliding on Split Graphs}},
  booktitle =	{36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
  pages =	{13:1--13:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-100-9},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{126},
  editor =	{Niedermeier, Rolf and Paul, Christophe},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.13},
  URN =		{urn:nbn:de:0030-drops-102529},
  doi =		{10.4230/LIPIcs.STACS.2019.13},
  annote =	{Keywords: reconfiguration, independent set, split graph}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2018.67
New Results on Directed Edge Dominating Set

Authors: Rémy Belmonte, Tesshu Hanaka, Ioannis Katsikarelis, Eun Jung Kim, and Michael Lampis

Published in: LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

Abstract
We study a family of generalizations of Edge Dominating Set on directed graphs called Directed (p,q)-Edge Dominating Set. In this problem an arc (u,v) is said to dominate itself, as well as all arcs which are at distance at most q from v, or at distance at most p to u. First, we give significantly improved FPT algorithms for the two most important cases of the problem, (0,1)-dEDS and (1,1)-dEDS (that correspond to versions of Dominating Set on line graphs), as well as polynomial kernels. We also improve the best-known approximation for these cases from logarithmic to constant. In addition, we show that (p,q)-dEDS is FPT parameterized by p+q+tw, but W-hard parameterized just by tw, where tw is the treewidth of the underlying graph of the input. We then go on to focus on the complexity of the problem on tournaments. Here, we provide a complete classification for every possible fixed value of p,q, which shows that the problem exhibits a surprising behavior, including cases which are in P; cases which are solvable in quasi-polynomial time but not in P; and a single case (p=q=1) which is NP-hard (under randomized reductions) and cannot be solved in sub-exponential time, under standard assumptions.
Cite as

Rémy Belmonte, Tesshu Hanaka, Ioannis Katsikarelis, Eun Jung Kim, and Michael Lampis. New Results on Directed Edge Dominating Set. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 67:1-67:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{belmonte_et_al:LIPIcs.MFCS.2018.67,
  author =	{Belmonte, R\'{e}my and Hanaka, Tesshu and Katsikarelis, Ioannis and Kim, Eun Jung and Lampis, Michael},
  title =	{{New Results on Directed Edge Dominating Set}},
  booktitle =	{43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)},
  pages =	{67:1--67:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-086-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{117},
  editor =	{Potapov, Igor and Spirakis, Paul and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.67},
  URN =		{urn:nbn:de:0030-drops-96490},
  doi =		{10.4230/LIPIcs.MFCS.2018.67},
  annote =	{Keywords: Edge Dominating Set, Tournaments, Treewidth}
}
Document
DOI: 10.4230/LIPIcs.FUN.2018.5
How Bad is the Freedom to Flood-It?

Authors: Rémy Belmonte, Mehdi Khosravian Ghadikolaei, Masashi Kiyomi, Michael Lampis, and Yota Otachi

Published in: LIPIcs, Volume 100, 9th International Conference on Fun with Algorithms (FUN 2018)

Abstract
Fixed-Flood-It and Free-Flood-It are combinatorial problems on graphs that generalize a very popular puzzle called Flood-It. Both problems consist of recoloring moves whose goal is to produce a monochromatic ("flooded") graph as quickly as possible. Their difference is that in Free-Flood-It the player has the additional freedom of choosing the vertex to play in each move. In this paper, we investigate how this freedom affects the complexity of the problem. It turns out that the freedom is bad in some sense. We show that some cases trivially solvable for Fixed-Flood-It become intractable for Free-Flood-It. We also show that some tractable cases for Fixed-Flood-It are still tractable for Free-Flood-It but need considerably more involved arguments. We finally present some combinatorial properties connecting or separating the two problems. In particular, we show that the length of an optimal solution for Fixed-Flood-It is always at most twice that of Free-Flood-It, and this is tight.
Cite as

Rémy Belmonte, Mehdi Khosravian Ghadikolaei, Masashi Kiyomi, Michael Lampis, and Yota Otachi. How Bad is the Freedom to Flood-It?. In 9th International Conference on Fun with Algorithms (FUN 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 100, pp. 5:1-5:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{belmonte_et_al:LIPIcs.FUN.2018.5,
  author =	{Belmonte, R\'{e}my and Khosravian Ghadikolaei, Mehdi and Kiyomi, Masashi and Lampis, Michael and Otachi, Yota},
  title =	{{How Bad is the Freedom to Flood-It?}},
  booktitle =	{9th International Conference on Fun with Algorithms (FUN 2018)},
  pages =	{5:1--5:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-067-5},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{100},
  editor =	{Ito, Hiro and Leonardi, Stefano and Pagli, Linda and Prencipe, Giuseppe},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2018.5},
  URN =		{urn:nbn:de:0030-drops-87961},
  doi =		{10.4230/LIPIcs.FUN.2018.5},
  annote =	{Keywords: flood-filling game, parameterized complexity}
}
Document
DOI: 10.4230/LIPIcs.STACS.2018.10
Parameterized (Approximate) Defective Coloring

Authors: Rémy Belmonte, Michael Lampis, and Valia Mitsou

Published in: LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

Abstract
In Defective Coloring we are given a graph G=(V,E) and two integers chi_d,Delta^* and are asked if we can partition V into chi_d color classes, so that each class induces a graph of maximum degree Delta^*. We investigate the complexity of this generalization of Coloring with respect to several well-studied graph parameters, and show that the problem is W-hard parameterized by treewidth, pathwidth, tree-depth, or feedback vertex set, if chi_d=2. As expected, this hardness can be extended to larger values of chi_d for most of these parameters, with one surprising exception: we show that the problem is FPT parameterized by feedback vertex set for any chi_d != 2, and hence 2-coloring is the only hard case for this parameter. In addition to the above, we give an ETH-based lower bound for treewidth and pathwidth, showing that no algorithm can solve the problem in n^{o(pw)}, essentially matching the complexity of an algorithm obtained with standard techniques. We complement these results by considering the problem's approximability and show that, with respect to Delta^*, the problem admits an algorithm which for any epsilon>0 runs in time (tw/epsilon)^{O(tw)} and returns a solution with exactly the desired number of colors that approximates the optimal Delta^* within (1+epsilon). We also give a (tw)^{O(tw)} algorithm which achieves the desired Delta^* exactly while 2-approximating the minimum value of chi_d. We show that this is close to optimal, by establishing that no FPT algorithm can (under standard assumptions) achieve a better than 3/2-approximation to chi_d, even when an extra constant additive error is also allowed.
Cite as

Rémy Belmonte, Michael Lampis, and Valia Mitsou. Parameterized (Approximate) Defective Coloring. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{belmonte_et_al:LIPIcs.STACS.2018.10,
  author =	{Belmonte, R\'{e}my and Lampis, Michael and Mitsou, Valia},
  title =	{{Parameterized (Approximate) Defective Coloring}},
  booktitle =	{35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
  pages =	{10:1--10:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-062-0},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{96},
  editor =	{Niedermeier, Rolf and Vall\'{e}e, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.10},
  URN =		{urn:nbn:de:0030-drops-85304},
  doi =		{10.4230/LIPIcs.STACS.2018.10},
  annote =	{Keywords: Treewidth, Parameterized Complexity, Approximation, Coloring}
}
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