Positional Games and QBF: The Corrective Encoding

Positional games are a mathematical class of two-player games comprising Tic-tac-toe and its generalizations. We propose a novel encoding of these games into Quantified Boolean Formulas (QBF) such that a game instance admits a winning strategy for first player if and only if the corresponding formula is true. Our approach improves over previous QBF encodings of games in multiple ways. First, it is generic and lets us encode other positional games, such as Hex. Second, structural properties of positional games together with a careful treatment of illegal moves let us generate more compact instances that can be solved faster by state-of-the-art QBF solvers. We establish the latter fact through extensive experiments. Finally, the compactness of our new encoding makes it feasible to translate realistic game problems. We identify a few such problems of historical significance and put them forward to the QBF community as milestones of increasing difficulty.Comment: Accepted for publication in the 23rd International Conference on Theory and Applications of Satisfiability Testing (SAT2020

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

The Maker-Maker domination game in forests

We study the Maker-Maker version of the domination game introduced in 2018 by Duch\^ene et al. Given a graph, two players alternately claim vertices. The first player to claim a dominating set of the graph wins. As the Maker-Breaker version, this game is PSPACE-complete on split and bipartite graphs. Our main result is a linear time algorithm to solve this game in forests. We also give a characterization of the cycles where the first player has a winning strategy

Apprendre à jouer aux jeux à deux joueurs à information parfaite sans connaissance

International audienceIn this paper, several techniques for learning game states evaluation functions by reinforcement are proposed. The first is to learn the values of the game tree instead of restricting oneself to the value of the root. The second is to replace the classic gain of a game (+1 / −1) with a heuris-tic favoring quick wins and slow defeats. The third corrects some evaluation functions taking into account the resolution of states. The fourth is a new action selection distribution. Finally, the fifth is a modification of the minimax with unbounded depth extending the best sequences of actions to the terminal states. In addition, we propose another variant of the unbounded minimax, which plays the safest action instead of playing the best action. The experiments conducted suggest that this improves the level of play during confrontations. Finally, we apply these different techniques to design a program-player to the Hex game (size 11) reaching the level of Mohex 2.0 with reinforcement learning from self-play without knowledge.Dans cet article, plusieurs techniques pour l'apprentissage par renforcement de fonctions d'évaluation d'états de jeu sont proposées. La première consiste à apprendre les va-leurs de l'arbre de jeu au lieu de se restreindre à la va-leur de la racine. La seconde consiste à remplacer le gain classique d'un jeu (+1 / −1) par une heuristique favo-risant les victoires rapides et les défaites lentes. La troi-sième permet de corriger certaines fonctions d'évaluation en tenant compte de la résolution des états. La quatrième est une nouvelle distribution de sélection d'actions. Enfin, la cinquième est une modification du minimax à profon-deur non bornée étendant les meilleures séquences d'ac-tions jusqu'aux états terminaux. En outre, nous proposons une autre variante du minimax non borné, qui joue l'ac-tion la plus sure au lieu de jouer la meilleure action. Les expériences menées suggèrent que cela améliore le niveau de jeux lors des confrontations. Enfin, nous appliquons ces différentes techniques pour concevoir un programme-joueur au jeu de Hex (taille 11) atteignant le niveau de Mohex 2.0 à la suite d'un apprentissage par renforcement contre soi-même sans utilisation de connaissance

The Largest Connected Subgraph Game

This paper introduces the largest connected subgraph game played on an undirected graph $G$. In each round, Alice first colours an uncoloured vertex of $G$ red, and then, Bob colours an uncoloured vertex of $G$ blue, with all vertices initially uncoloured. Once all the vertices are coloured, Alice (Bob, resp.) wins if there is a red (blue, resp.) connected subgraph whose order is greater than the order of any blue (red, resp.) connected subgraph. We first prove that Bob can never win, and define a large class of graphs (called reflection graphs) in which the game is a draw. We then show that determining the outcome of the game is PSPACE-complete, even in bipartite graphs of small diameter, and that recognising reflection graphs is GI-hard. We also prove that the game is a draw in paths if and only if the path is of even order or has at least $11$ vertices, and that Alice wins in cycles if and if only if the cycle is of odd length. Lastly, we give an algorithm to determine the outcome of the game in cographs in linear time

On the complexity of connection games

In this paper, we study three connection games among the most widely played: Havannah, Twixt, and Slither. We show that determining the outcome of an arbitrary input position is PSPACE-complete in all three cases. Our reductions are based on the popular graph problem Generalized Geography and on Hex itself. We also consider the complexity of generalizations of Hex parameterized by the length of the solution and establish that while Short Generalized Hex is W[1]-hard, Short Hex is FPT. Finally, we prove that the ultra-weak solution to the empty starting position in hex cannot be fully adapted to any of these three games.Comment: Subsumes and extends https://arxiv.org/abs/1403.6518 significantl