On the computational complexity of a game of cops and robbers

We study the computational complexity of a perfect-information two-player game proposed by Aigner and Fromme (1984) [1]. The game takes place on an undirected graph where n simultaneously moving cops attempt to capture a single robber, all moving at the same speed. The players are allowed to pick their starting positions at the first move. The question of the computational complexity of deciding this game was raised by Goldstein and Reingold (1995) [9]. We prove that the game is hard for PSPACE.©2012 Elsevier B.V. All rights reserved

Spy-Game on graphs

International audienceWe 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 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 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 > 0 such that Ω(n 1+) guards are required to win in any n × n grid

To satisfy impatient Web surfers is hard

International audiencePrefetching is a basic mechanism for faster data access and efficient computing. An important issue in prefetching is the tradeoff between the amount of network's resources wasted by the prefetching and the gain of time. For instance, in the Web, browsers may download documents in advance while a Web surfer is surfing. Since the Web surfer follows the hyperlinks in an unpredictable way, the choice of the Web pages to be prefetched must be computed online. The question is then to determine the minimum amount of resources used by prefetching that ensures that all documents accessed by theWeb surfer have previously been loaded in the cache. We model this problem as a two-player game similar to Cops and Robber Games in graphs. Let k 1 be any integer. The first player, a fugitive, starts on a marked vertex of a (di)graph G. The second player, an observer, marks at most k vertices, then the fugitive moves along one edge/arc of G to a new vertex, then the observer marks at most k vertices, etc. The fugitive wins if it enters an unmarked vertex, and the observer wins otherwise. The surveillance number of a (di)graph is the minimum k such that the observer marking at most k vertices at each step can win against any strategy of the fugitive. We also consider the connected variant of this game, i.e., when a vertex can be marked only if it is adjacent to an already marked vertex. We study the computational complexity of the game. All our results hold for both variants, connected or unrestricted. We show that deciding whether the surveillance number of a chordal graph is at most 2 is NP-hard. We also prove that deciding if the surveillance number of a DAG is at most 4 is PSPACEcomplete. Moreover, we show that the problem of computing the surveillance number is NP-hard in split graphs. On the other hand, we provide polynomial time algorithms computing surveillance numbers of trees and interval graphs. Moreover, in the case of trees, we establish a combinatorial characterization of the surveillance number

Straight Line Movement in Morphing and Pursuit Evasion

Piece-wise linear structures are widely used to define problems and to represent simplified solutions in computational geometry. A piece-wise linear structure consists of straight-line or linear pieces connected together in a continuous geometric environment like 2D or 3D Euclidean spaces. In this thesis two different problems both with the approach of finding piece-wise linear solutions in 2D space are defined and studied: straight-line pursuit evasion and straight-line morphing. Straight-line pursuit evasion is a geometric version of the famous cops and robbers game that is defined in this thesis for the first time. The game is played in a simply connected region in 2D. It is a full information game where the players take turns. The cop’s goal is to catch the robber. In a turn, each player may move any distance along a straight line as long as the line segment connecting their current location to the new location is not blocked by the region’s boundary. We first prove that the cop can always win the game when the players move on the visibility graph of a simple polygon. We prove this by showing that the visibility graph of a simple polygon is “dismantlable” (the known class of cop-win graphs). Polygon visibility graphs are also shown to be 2-dismantlable. Two other settings of the game are also studied in this thesis: when the players are free to move on the infinitely many points inside a simple polygon, and inside a splinegon. In both cases we show that the cop can always win the game. For the case of polygons, the proposed cop strategy gives an asymptotically tight linear bound on the number of steps the cop needs to catch the robber. For the case of splinegons, the cop may need a quadratic number of steps with the proposed strategy, while our best lower bound is linear. Straight-line morphing is a type of morphing first defined in this thesis that provides a nice and smooth transformation between straight-line graph drawings in 2D. In straight- line morphing, each vertex of the graph moves forward along the line segment connecting its initial position to its final position. The vertex trajectories in straight-line morphing are very simple, but because the speed of each vertex may vary, straight-line morphs are more general than the commonly used “linear morphs” where each vertex moves at uniform speed. We explore the problem of whether an initial planar straight-line drawing of a graph can be morphed to a final straight-line drawing of the graph using a straight-line morph that preserves planarity at all times. We prove that this problem is NP-hard even for the special case where the graph drawing consists of disjoint segments. We then look at some restricted versions of the straight-line morphing: when only one vertex moves at a time, when the vertices move one by one to their final positions uninterruptedly, and when the edges morph one by one to their final configurations in the case of disjoint segments. Some of the variations are shown to be still NP-complete while some others are solvable in polynomial time. We conjecture that the class of planar straight-line morphs is as powerful as the class of planar piece-wise linear straight-line morphs. We also explore a simpler problem where for each edge the quadrilateral formed by its initial and final positions together with the trajectories of its two vertices is convex. There is a necessary condition for this case that we conjecture is also sufficient for paths and cycles

Prospectus, August 12, 2009

https://spark.parkland.edu/prospectus_2009/1019/thumbnail.jp

Multi-agent algorithms with assignment strategy pursuing multiple moving targets in dynamic environments

Devising intelligent agents to successfully plan a path to a target is a common problem in artificial intelligence and in recent years, attention has increased to multi-agent pathfinding problems, especially due to the expansion in computer video games and robotics. Pathfinding for agents in real-world applications is a defined problem of multi-agent systems, where pursuing agents collaborate among themselves and autonomously plan their path to the targets. There are multi-agent algorithms that provide solutions with the shortest path without considering other pursuers and several of those use coordination. However, less attention has been paid to computing an assignment strategy for the pursuers and finding paths that collectively surround the targets. Comparatively fewer studies have been on target algorithms either. Besides, the multi-agent pathfinding problem becomes even more challenging if the goal destinations change over time. Existing solutions consider either a single target with moving capability or multiple targets that are stationary. The work presented in this thesis considers multiple moving targets in multi-agent systems. Therefore, the path planning problem for multiple pursuing agents requires more efficient pathfinding algorithms. In addition, when the target algorithms are improved for advanced behaviour with moving capabilities that smartly evade the pursuers makes the problem even harder. The research reported in this thesis aims to investigate multi-agent search algorithms to address the challenge associated with pursuing agents towards moving targets within a dynamically changing environment. In multi-agent scenarios, agents compute a path towards the target, while these target destinations in some cases are predefined in advance. Thus, this research proposes to investigate a solution to the path planning problem by utilising heuristic algorithms as well as assignment strategies for multiple pursuing agents. Furthermore, a state-of-the-art moving target algorithm, TrailMax, has been enhanced and implemented for multiple agent pathfinding problems, which aims to maximise the capture time if possible until timeout. The focus of this thesis is the investigation of the assignment strategy algorithms to coordinate multiple pursuing agents and explore pathfinding search algorithms to find a route towards moving targets. This will be achieved by dividing it into two stages. The first one is the coupled approach where the assignment strategy with a given criterion finds the optimal combination based on the current position of players. The second stage is the decoupled approach, where each agent independently finds its path towards the moving target. On the other hand, targets flee from pursuing agents using the specified escaping strategy. The novel contributions of the research presented in this thesis are summarised as follows: - A new algorithm is developed that uses existing assignment strategies, sum-of-costs and makespan, to assign targets, and then runs repetitive A* search until reaches the target. - An enhancement is provided for a state-of-the-art target algorithm that takes smart moves by avoiding capture from all pursuers. - To improve efficiency, six new approaches are investigated to find an optimal agent-to-target combination for target assignment. - A novel multi-agent algorithm is developed which uses cover heuristics to maximise its coverage to outmanoeuvre, trap and catch moving targets. The proposed pathfinding solutions and the results presented in this thesis demonstrate a significant contribution towards search algorithms in multi-agent systems

The Other Side of Fun

The Other Side of Fun is a collection of creative non-fiction essays that examine the relationship between several cultural pastimes and our society as a whole. The thoughts, feelings, and observations made throughout these essays are reflections of my time spent working various jobs pertaining to some form of entertainment. Mayflies explores my time as a game-day security worker for the Cleveland Indians, examining the relationship between unionized labor and the lifestyle that encompasses it. Spiders chronicles my time spent as a Resident Assistant at Cleveland State, investigating the deep web and the potential dangers that technology can bring. House Rules details my experiences at the Jack Casino, exploring society\u27s obsession with wealth. Ghosts looks at society\u27s use of tradition, documenting an evening spent working as a bouncer at one of the busiest bars the night before Halloween. Last is Cutting Weight , an essay that discusses the world of organized cage-fighting and the impact it has had on both our culture and my own life. These essays serve as a critique to the way our world operates; a collection of observations that look to challenge the reader\u27s perception of our societal ideologies and values

Albuquerque Morning Journal, 11-27-1921

https://digitalrepository.unm.edu/abq_mj_news/1397/thumbnail.jp