Pursuing a fast robber on a graph

Fomin, Fedor V.; Golovach, Petr A.; Kratochvíl, Jan; Nisse, Nicolas; Suchan, Karol

doi:10.1016/j.tcs.2009.12.010

Cited by 66 publications

(32 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Thus, various variations of the game have been studied [2,7,11,20,23]. The approach chosen by Fomin, Golovach, Kratochvíl, Nisse, and Suchan [18] is to allow the robber to move faster than the cops. Inspired by their work, in this article we let the robber take any path from her current position in her turn, but she is not allowed to pass through a vertex occupied by a cop.…”

Section: Introductionmentioning

confidence: 99%

Chasing a Fast Robber on Planar Graphs and Random Graphs

Alon

Mehrabian

2014

J. Graph Theory

View full text Add to dashboard Cite

We consider a variant of the Cops and Robber game, in which the robber has unbounded speed, i.e., can take any path from her vertex in her turn, but she is not allowed to pass through a vertex occupied by a cop. Let c ∞ (G) denote the number of cops needed to capture the robber in a graph G in this variant, and let tw(G) denote the treewidth of G. We show that if G is planar then c ∞ (G) = Θ(tw(G)), and there is a constant-factor approximation algorithm for computing c ∞ (G). We also determine, up to constant factors, the value of c ∞ (G) of the Erdős-Rényi random graph G = G(n, p) for all admissible values of p, and show that when the average degree is ω(1), c ∞ (G) is typically asymptotic to the domination number.

show abstract

Section: Introductionmentioning

confidence: 99%

Chasing a Fast Robber on Planar Graphs and Random Graphs

Alon

Mehrabian

2014

J. Graph Theory

View full text Add to dashboard Cite

show abstract

“…For instance, the exact number of cops with speed one required to capture a robber with speed two is unknown in 2-dimensional grids [11,5]. One of our hopes when introducing the Spy-game is that it will lead us to a new approach to tackle this problem.…”

Section: Related Workmentioning

confidence: 99%

Spy-game on graphs: Complexity and simple topologies

Cohen¹,

Martins²,

Inerney³

et al. 2018

Theoretical Computer Science

Self Cite

View full text Add to dashboard Cite

We define and study the following two-player game on a graph G. Let k ∈ N * . A set of k guards is occupying some vertices of G while one spy is standing at some node. At each turn, first the spy may move along at most s edges, where s ∈ N * is his speed. Then, each guard may move along one edge. The spy and the guards may occupy the same vertices. The spy has to escape the surveillance of the guards, i.e., must reach a vertex at distance more than d ∈ N (a predefined distance) from every guard. Can the spy win against k guards? Similarly, what is the minimum distance d such that k guards may ensure that at least one of them remains at distance at most d from the spy? This game generalizes two well-studied games: Cops and robber games (when s = 1) and Eternal Dominating Set (when s is unbounded).We consider the computational complexity of the problem, showing that it is NP-hard (for every speed s and distance d) and that some variant of it is PSPACE-hard in DAGs. Then, we establish tight tradeoffs between the number of guards, the speed s of the spy and the required distance d when G is a path or a cycle.

show abstract

“…These combinatorial problems are generally "hard": Cops and Robber games are EXPTIME-complete [18] and W [2]-hard [12] and the Surveillance game is NP-hard and even PSPACE-complete in directed graphs [11]. Moreover, many longstanding (probably difficult) open questions have not been solved yet for these games.…”

Section: Introductionmentioning

confidence: 99%

“…A classical approach to tackle these open problems has been to study variants of these games in which one of the players often has some restrictions. For instance, the robber may be faster than the cops [2,12], the cops may capture at some distance [4], the surveyed area may be forced to be connected [13], etc. Another approach may be to restrict the games to particular graph classes such as trees [19], grids [16,21], planar graphs [1], bounded treewidth graphs [17,20], etc.…”

Section: Introductionmentioning

confidence: 99%

Study of a Combinatorial Game in Graphs Through Linear Programming

et al. 2018

Self Cite

View full text Add to dashboard Cite

In the Spy Game played on a graph G, a single spy travels the vertices of G at speed s, while multiple slow guards strive to have, at all times, one of them within distance d of that spy. In order to determine the smallest number of guards necessary for this task, we analyze the game through a Linear Programming formulation and the fractional strategies it yields for the guards. We then show the equivalence of fractional and integral strategies in trees. This allows us to design a polynomial-time algorithm for computing an optimal strategy in this class of graphs. Using duality in Linear Programming, we also provide non-trivial bounds on the fractional guard-number of grids and torus which gives a lower bound for the integral guardnumber in these graphs. We believe that the approach using fractional relaxation and Linear Programming is promising to obtain new results in the field of combinatorial games.

show abstract

Pursuing a fast robber on a graph

Cited by 66 publications

References 30 publications

Chasing a Fast Robber on Planar Graphs and Random Graphs

Chasing a Fast Robber on Planar Graphs and Random Graphs

Spy-game on graphs: Complexity and simple topologies

Study of a Combinatorial Game in Graphs Through Linear Programming

Contact Info

Product

Resources

About