Pursuing a fast robber on a graph

AbstractThe Cops and Robbers game as originally defined independently by Quilliot and by Nowakowski and Winkler in the 1980s has been much studied, but very few results pertain to the algorithmic and complexity aspects of it. In this paper we prove that computing the minimum number of cops that are guaranteed to catch a robber on a given graph is NP-hard and that the parameterized version of the problem is W[2]-hard; the proof extends to the case where the robber moves s time faster than the cops. We show that on split graphs, the problem is polynomially solvable if s=1 but is NP-hard if s=2. We further prove that on graphs of bounded cliquewidth the problem is polynomially solvable for s≤2. Finally, we show that for planar graphs the minimum number of cops is unbounded if the robber is faster than the cops

Recontamination Helps a Lot to Hunt a Rabbit

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

Current Algorithms for Detecting Subgraphs of Bounded Treewidth are Probably Optimal

The Subgraph Isomorphism problem is of considerable importance in computer science. We examine the problem when the pattern graph H is of bounded treewidth, as occurs in a variety of applications. This problem has a well-known algorithm via color-coding that runs in time $O(n^{tw(H)+1})$ [Alon, Yuster, Zwick'95], where $n$ is the number of vertices of the host graph $G$. While there are pattern graphs known for which Subgraph Isomorphism can be solved in an improved running time of $O(n^{tw(H)+1-\varepsilon})$ or even faster (e.g. for $k$-cliques), it is not known whether such improvements are possible for all patterns. The only known lower bound rules out time $n^{o(tw(H) / \log(tw(H)))}$ for any class of patterns of unbounded treewidth assuming the Exponential Time Hypothesis [Marx'07]. In this paper, we demonstrate the existence of maximally hard pattern graphs $H$ that require time $n^{tw(H)+1-o(1)}$. Specifically, under the Strong Exponential Time Hypothesis (SETH), a standard assumption from fine-grained complexity theory, we prove the following asymptotic statement for large treewidth $t$: For any $\varepsilon > 0$ there exists $t \ge 3$ and a pattern graph $H$ of treewidth $t$ such that Subgraph Isomorphism on pattern $H$ has no algorithm running in time $O(n^{t+1-\varepsilon})$. Under the more recent 3-uniform Hyperclique hypothesis, we even obtain tight lower bounds for each specific treewidth $t \ge 3$: For any $t \ge 3$ there exists a pattern graph $H$ of treewidth $t$ such that for any $\varepsilon>0$ Subgraph Isomorphism on pattern $H$ has no algorithm running in time $O(n^{t+1-\varepsilon})$. In addition to these main results, we explore (1) colored and uncolored problem variants (and why they are equivalent for most cases), (2) Subgraph Isomorphism for $tw < 3$, (3) Subgraph Isomorphism parameterized by pathwidth, and (4) a weighted problem variant

Geometric Pursuit Evasion

In this dissertation we investigate pursuit evasion problems set in geometric environments. These games model a variety of adversarial situations in which a team of agents, called pursuers, attempts to catch a rogue agent, called the evader. In particular, we consider the following problem: how many pursuers, each with the same maximum speed as the evader, are needed to guarantee a successful capture? Our primary focus is to provide combinatorial bounds on the number of pursuers that are necessary and sufficient to guarantee capture. The first problem we consider consists of an unpredictable evader that is free to move around a polygonal environment of arbitrary complexity. We assume that the pursuers have complete knowledge of the evader's location at all times, possibly obtained through a network of cameras placed in the environment. We show that regardless of the number of vertices and obstacles in the polygonal environment, three pursuers are always sufficient and sometimes necessary to capture the evader. We then consider several extensions of this problem to more complex environments. In particular, suppose the players move on the surface of a 3-dimensional polyhedral body; how many pursuers are required to capture the evader? We show that 4 pursuers always suffice (upper bound), and that 3 are sometimes necessary (lower bound), for any polyhedral surface with genus zero. Generalizing this bound to surfaces of genus g, we prove the sufficiency of (4g + 4) pursuers. Finally, we show that 4 pursuers also suffice under the "weighted region" constraints, where the movement costs through different regions of the (genus zero) surface have (different) multiplicative weights. Next we consider a more general problem with a less restrictive sensing model. The pursuers' sensors are visibility based, only providing the location of the evader if it is in direct line of sight. We begin my making only the minimalist assumption that pursuers and the evader have the same maximum speed. When the environment is a simply-connected (hole-free) polygon of n vertices, we show that Θ(n^1/2 ) pursuers are both necessary and sufficient in the worst-case. When the environment is a polygon with holes, we prove a lower bound of Ω(n^2/3 ) and an upper bound of O(n^5/6 ) pursuers, where n includes the vertices of the hole boundaries. However, we show that with realistic constraints on the polygonal environment these bounds can be drastically improved. Namely, if the players' movement speed is small compared to the features of the environment, we give an algorithm with a worst case upper bound of O(log n) pursuers for simply-connected n-gons and O(√h + log n) for polygons with h holes. The final problem we consider takes a small step toward addressing the fact that location sensing is noisy and imprecise in practice. Suppose a tracking agent wants to follow a moving target in the two-dimensional plane. We investigate what is the tracker's best strategy to follow the target and at what rate does the distance between the tracker and target grow under worst-case localization noise. We adopt a simple but realistic model of relative error in sensing noise: the localization error is proportional to the true distance between the tracker and the target. Under this model we are able to give tight upper and lower bounds for the worst-case tracking performance, both with or without obstacles in the Euclidean plane

Justice, Liability, and Blame: Community Views and the Criminal Law

This book reports empirical studies on 18 different areas of substantive criminal law in which the study results showing ordinary people’s judgments of justice are compared to the governing legal doctrine to highlight points of agreement and disagreement. The book also identifies trends and patterns in agreement and disagreement and discusses the implications for the formulation of criminal law. The chapters include: Chapter 1. Community Views and the Criminal Law (Introduction; An Overview; Why Community Views Should Matter; Research Methods) Chapter 2. Doctrines of Criminalization: What Conduct Should Be Criminal? (Objective Requirements of Attempt (Study 1); Creating a Criminal Risk (Study 2); Objective Requirements of Complicity (Study 3); Omission Liability (Study 4); Chapter Summary) Chapter 3. Doctrines of Justification: When Should It Be Lawful to Engage in Conduct That Normally Would Constitute a Violation? (Use of Deadly Force in Self-Defense (Study 5); Use of Force in Defense of Property (Study 6); Citizens\u27 Law Enforcement Authority (Study 7); Chapter Summary) Chapter 4. Doctrines of Culpability: When Is Violation of a Legal Rule Blameworthy? (Offense Culpability Requirements and Mistake/Accident Defenses (Study 8); Culpability Requirements for Complicity (Study 9); Voluntary Intoxication (Study 10); Individualization of the Objective Standard of Negligence (Study 11); Chapter Summary) Chapter 5. Doctrines of Excuse: When Is a Rule Violation Blameless? (Insanity (Study 12); Immaturity and Involuntary Intoxication (Study 13); Duress and Entrapment Defenses (Study 14); Chapter Summary) Chapter 6. Doctrines of Grading: What Degree of Punishment Is Deserved for a Blameworthy Violation? (The Seriousness of the Offense: Sexual Offenses (Study 15); The Culpability of the Person: Felony Murder (Study 16); The Strength of the Person\u27s Connection with the Prohibited Result: Causation Requirements (Study 17); Punishment for Multiple Offenses (Study 18); Chapter Summary) Chapter 7. Community Views and Criminal Codes Conflict (When Code and Community Agree; When Code and Community Disagree; Liability Requirements Versus Liability Factors and Dichotomous Functions Versus Continuous Functions; Criminal Liability Without Punishment; The Jury as a Resolver of Code-Community Conflicts