DROPS

Document

DOI: 10.4230/LIPIcs.MFCS.2023.42

Recontamination Helps a Lot to Hunt a Rabbit

Authors: Thomas Dissaux, Foivos Fioravantes, Harmender Gahlawat, and Nicolas Nisse

Published in: LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

Abstract

The Hunters and Rabbit game is played on a graph G where the Hunter player shoots at k vertices in every round while the Rabbit player occupies an unknown vertex and, if it is not shot, must move to a neighbouring vertex after each round. The Rabbit player wins if it can ensure that its position is never shot. The Hunter player wins otherwise. The hunter number h(G) of a graph G is the minimum integer k such that the Hunter player has a winning strategy (i.e., allowing him to win whatever be the strategy of the Rabbit player). This game has been studied in several graph classes, in particular in bipartite graphs (grids, trees, hypercubes...), but the computational complexity of computing h(G) remains open in general graphs and even in more restricted graph classes such as trees. To progress further in this study, we propose a notion of monotonicity (a well-studied and useful property in classical pursuit-evasion games such as Graph Searching games) for the Hunters and Rabbit game imposing that, roughly, a vertex that has already been shot "must not host the rabbit anymore". This allows us to obtain new results in various graph classes. More precisely, let the monotone hunter number mh(G) of a graph G be the minimum integer k such that the Hunter player has a monotone winning strategy. We show that pw(G) ≤ mh(G) ≤ pw(G)+1 for any graph G with pathwidth pw(G), which implies that computing mh(G), or even approximating mh(G) up to an additive constant, is NP-hard. Then, we show that mh(G) can be computed in polynomial time in split graphs, interval graphs, cographs and trees. These results go through structural characterisations which allow us to relate the monotone hunter number with the pathwidth in some of these graph classes. In all cases, this allows us to specify the hunter number or to show that there may be an arbitrary gap between h and mh, i.e., that monotonicity does not help. In particular, we show that, for every k ≥ 3, there exists a tree T with h(T) = 2 and mh(T) = k. We conclude by proving that computing h (resp., mh) is FPT parameterised by the minimum size of a vertex cover.

Cite as

Thomas Dissaux, Foivos Fioravantes, Harmender Gahlawat, and Nicolas Nisse. Recontamination Helps a Lot to Hunt a Rabbit. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 42:1-42:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{dissaux_et_al:LIPIcs.MFCS.2023.42,
  author =	{Dissaux, Thomas and Fioravantes, Foivos and Gahlawat, Harmender and Nisse, Nicolas},
  title =	{{Recontamination Helps a Lot to Hunt a Rabbit}},
  booktitle =	{48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
  pages =	{42:1--42:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-292-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{272},
  editor =	{Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.42},
  URN =		{urn:nbn:de:0030-drops-185763},
  doi =		{10.4230/LIPIcs.MFCS.2023.42},
  annote =	{Keywords: Hunter and Rabbit, Monotonicity, Graph Searching}
}

Document

DOI: 10.4230/LIPIcs.SWAT.2022.24

Complexity of Finding Maximum Locally Irregular Induced Subgraphs

Authors: Foivos Fioravantes, Nikolaos Melissinos, and Theofilos Triommatis

Published in: LIPIcs, Volume 227, 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)

Abstract

If a graph G is such that no two adjacent vertices of G have the same degree, we say that G is locally irregular. In this work we introduce and study the problem of identifying a largest induced subgraph of a given graph G that is locally irregular. Equivalently, given a graph G, find a subset S of V(G) with minimum order, such that by deleting the vertices of S from G results in a locally irregular graph; we denote with I(G) the order of such a set S. We first examine some easy graph families, namely paths, cycles, trees, complete bipartite and complete graphs. However, we show that the decision version of the introduced problem is NP-Complete, even for restricted families of graphs, such as subcubic planar bipartite, or cubic bipartite graphs. We then show that we can not even approximate an optimal solution within a ratio of 𝒪(n^{1-1/k}), where k ≥ 1 and n is the order the graph, unless 𝒫=NP, even when the input graph is bipartite. Then, looking for more positive results, we turn our attention towards computing I(G) through the lens of parameterised complexity. In particular, we provide two algorithms that compute I(G), each one considering different parameters. The first one considers the size of the solution k and the maximum degree Δ of G with running time (2Δ)^kn^{𝒪(1)}, while the second one considers the treewidth tw and Δ of G, and has running time Δ^{2tw}n^{𝒪(1)}. Therefore, we show that the problem is FPT by both k and tw if the graph has bounded maximum degree Δ. Since these algorithms are not FPT for graphs with unbounded maximum degree (unless we consider Δ + k or Δ + tw as the parameter), it is natural to wonder if there exists an algorithm that does not include additional parameters (other than k or tw) in its dependency. We answer negatively, to this question, by showing that our algorithms are essentially optimal. In particular, we prove that there is no algorithm that computes I(G) with dependence f(k)n^{o(k)} or f(tw)n^{o(tw)}, unless the ETH fails.

Cite as

Foivos Fioravantes, Nikolaos Melissinos, and Theofilos Triommatis. Complexity of Finding Maximum Locally Irregular Induced Subgraphs. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 24:1-24:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{fioravantes_et_al:LIPIcs.SWAT.2022.24,
  author =	{Fioravantes, Foivos and Melissinos, Nikolaos and Triommatis, Theofilos},
  title =	{{Complexity of Finding Maximum Locally Irregular Induced Subgraphs}},
  booktitle =	{18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)},
  pages =	{24:1--24:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-236-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{227},
  editor =	{Czumaj, Artur and Xin, Qin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2022.24},
  URN =		{urn:nbn:de:0030-drops-161842},
  doi =		{10.4230/LIPIcs.SWAT.2022.24},
  annote =	{Keywords: Locally irregular, largest induced subgraph, FPT, treewidth, W-hardness, approximability}
}

Search Results

Documents authored by Fioravantes, Foivos

Recontamination Helps a Lot to Hunt a Rabbit

Abstract

Cite as

Complexity of Finding Maximum Locally Irregular Induced Subgraphs

Abstract

Cite as

Thanks for your feedback!

Could not send message