Recontamination helps a lot to hunt a rabbit

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21240%2F23%3A00368685" target="_blank" >RIV/68407700:21240/23:00368685 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.4230/LIPICS.MFCS.2023.42" target="_blank" >https://doi.org/10.4230/LIPICS.MFCS.2023.42</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.4230/LIPICS.MFCS.2023.42" target="_blank" >10.4230/LIPICS.MFCS.2023.42</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Recontamination helps a lot to hunt a rabbit

Popis výsledku v původním jazyce

The textsc{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 textsc{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 $pw(G) leq mh(G) leq pw(G)+1$ 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 textsf{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 $kgeq 3$, 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.

Název v anglickém jazyce

Recontamination helps a lot to hunt a rabbit

Popis výsledku anglicky

The textsc{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 textsc{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 $pw(G) leq mh(G) leq pw(G)+1$ 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 textsf{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 $kgeq 3$, 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.

Druh

D - Stať ve sborníku

CEP obor

—

OECD FORD obor

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2023

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název statě ve sborníku

Proceedings of the 48th International Symposium on Mathematical Foundations of Computer Science

ISBN

978-3-95977-292-1

ISSN

1868-8969

e-ISSN

—

Počet stran výsledku

14

Strana od-do

591-604

Název nakladatele

Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik

Místo vydání

Dagstuhl

Místo konání akce

Bordeaux

Datum konání akce

28. 8. 2023

Typ akce podle státní příslušnosti

WRD - Celosvětová akce

Kód UT WoS článku

—