Cops and Robbers is EXPTIME-complete

We investigate the computational complexity of deciding whether k cops can capture a robber on a graph G. In 1995, Goldstein and Reingold conjectured that the problem is EXPTIME-complete when both G and k are part of the input; we prove this conjecture.Comment: v2: updated figures and slightly clarified some minor point

Characterizations and algorithms for generalized Cops and Robbers games

We propose a definition of generalized Cops and Robbers games where there are two players, the Pursuer and the Evader, who each move via prescribed rules. If the Pursuer can ensure that the game enters into a fixed set of final positions, then the Pursuer wins; otherwise, the Evader wins. A relational characterization of the games where the Pursuer wins is provided. A precise formula is given for the length of the game, along with an algorithm for computing if the Pursuer has a winning strategy whose complexity is a function of the parameters of the game. For games where the position of one player does not affect the available moves of he other, a vertex elimination ordering characterization, analogous to a cop-win ordering, is given for when the Pursuer has a winning strategy

Graphs with Large Girth and Small Cop Number

In this paper we consider the cop number of graphs with no, or few, short cycles. We show that when the girth of $G$ is at least $8$ and the minimum degree is sufficiently large, $\delta \geq (\ln{n})^{\frac{1}{1-\alpha}}$ where $\alpha \in (0,1)$, then $c(G) = o(n \delta^{\beta -\lfloor \frac{g}{4} \rfloor})$ as $\delta \rightarrow \infty$ where $\beta> 1-\alpha$. This extends work of Frankl and implies that if $G$ is large and dense in the sense that $\delta \geq n^{\frac{2}{g} - o(1)}$ while also having girth $g \geq 8$, then $G$ satisfies Meyniel's conjecture, that is $c(G) = O(\sqrt{n})$. Moreover, it implies that if $G$ is large and dense in the sense that there $\delta \geq n^{\epsilon}$ for some $\epsilon >0$, while also having girth $g \geq 8$, then there exists an $\alpha>0$ such that $c(G) = O(n^{1-\alpha})$, thereby satisfying the weak Meyniel's conjecture. Of course, this implies similar results for dense graphs with small, that is $O(n^{1-\alpha})$, numbers of short cycles, as each cycle can be broken by adding a single cop. We also, show that there are graphs $G$ with girth $g$ and minimum degree $\delta$ such that the cop number is at most $o(g (\delta-1)^{(1+o(1))\frac{g}{4}})$. This resolves a recent conjecture by Bradshaw, Hosseini, Mohar, and Stacho, by showing that the constant $\frac{1}{4}$ cannot be improved in the exponent of a lower bound $c(G) \geq \frac{1}{g} (\delta - 1)^{\lfloor \frac{g-1}{4}\rfloor}$.Comment: 7 pages, 0 figures, 0 table

Fine-grained Lower Bounds on Cops and Robbers

Cops and Robbers is a classic pursuit-evasion game played between a group of g cops and one robber on an undirected N-vertex graph G. We prove that the complexity of deciding the winner in the game under optimal play requires Omega (N^{g-o(1)}) time on instances with O(N log^2 N) edges, conditioned on the Strong Exponential Time Hypothesis. Moreover, the problem of calculating the minimum number of cops needed to win the game is 2^{Omega (sqrt{N})}, conditioned on the weaker Exponential Time Hypothesis. Our conditional lower bound comes very close to a conditional upper bound: if Meyniel\u27s conjecture holds then the cop number can be decided in 2^{O(sqrt{N}log N)} time. In recent years, the Strong Exponential Time Hypothesis has been used to obtain many lower bounds on classic combinatorial problems, such as graph diameter, LCS, EDIT-DISTANCE, and REGEXP matching. To our knowledge, these are the first conditional (S)ETH-hard lower bounds on a strategic game

Spy-Game on Graphs

We define and study the following two-player game on a graph G. Let k in 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 in N^* is his speed. Then, each guard may move along one edge. The spy and the guards may occupy same vertices. The spy has to escape the surveillance of the guards, i.e., must reach a vertex at distance more than d in 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 and that it is PSPACE-hard in DAGs. Then, we establish tight tradeoffs between the number of guards and the required distance d when G is a path or a cycle. Our main result is that there exists beta>0 such that Omega(n^{1+beta}) guards are required to win in any n*n grid

Parameterized Analysis of the Cops and Robber Game

Pursuit-evasion games have been intensively studied for several decades due to their numerous applications in artificial intelligence, robot motion planning, database theory, distributed computing, and algorithmic theory. Cops and Robber (CnR) is one of the most well-known pursuit-evasion games played on graphs, where multiple cops pursue a single robber. The aim is to compute the cop number of a graph, k, which is the minimum number of cops that ensures the capture of the robber. From the viewpoint of parameterized complexity, CnR is W[2]-hard parameterized by k [Fomin et al., TCS, 2010]. Thus, we study structural parameters of the input graph. We begin with the vertex cover number (vcn). First, we establish that k ? vcn/3+1. Second, we prove that CnR parameterized by vcn is FPT by designing an exponential kernel. We complement this result by showing that it is unlikely for CnR parameterized by vcn to admit a polynomial compression. We extend our exponential kernels to the parameters cluster vertex deletion number and deletion to stars number, and design a linear vertex kernel for neighborhood diversity. Additionally, we extend all of our results to several well-studied variations of CnR

Bounds on the length of a game of Cops and Robbers

In the game of Cops and Robbers, a team of cops attempts to capture a robber on a graph G. All players occupy vertices of G. The game operates in rounds; in each round the cops move to neighboring vertices, after which the robber does the same. The minimum number of cops needed to guarantee capture of a robber on G is the cop number of G, denoted c(G), and the minimum number of rounds needed for them to do so is the capture time. It has long been known that the capture time of an n-vertex graph with cop number k is O(nk+1). More recently, Bonato, Golovach, Hahn, and Kratochvíl ([3], 2009) and Gavenčiak ([10], 2010) showed that for k = 1, this upper bound is not asymptotically tight: for graphs with cop number 1, the cop can always win within n − 4 rounds. In this paper, we show that the upper bound is tight when k ≥ 2: for fixed k ≥ 2, we construct arbitrarily large graphs G having capture time at least (|V (G)|/40k4 )k+1. In the process of proving our main result, we establish results that may be of independent interest. In particular, we show that the problem of deciding whether k cops can capture a robber on a directed graph is polynomial-time equivalent to deciding whether k cops can capture a robber on an undirected graph. As a corollary of this fact, we obtain a relatively short proof of a major conjecture of Goldstein and Reingold ([11], 1995), which was recently proved through other means ([12], 2015). We also show that n-vertex strongly-connected directed graphs with cop number 1 can have capture time Ω(n2), thereby showing that the result of Bonato et al. [3] does not extend to the directed setting