A Game-Theoretic Approach to Constraint Satisfaction

We shed light on the connections between different approaches to constraint satisfaction by showing that the main consistency concepts used to derive tractability results for constraint satisfaction are intimately related to certain combinatorial pebble games, called the existential -pebble games, that were originally introduced in the context of Datalog. The crucial insight relating pebble games to constraint satisfaction is that the key concept of strong -consistency is equivalent to a condition on winning strategies for the Duplicator player in the existential -pebble game. We use this insight to show that strong -consistency can be established if and only if the Duplicator wins the existential -pebble game. Moreover, whenever strong -consistency can be established, one method for doing this is to first compute the largest winning strategy for the Duplicator in the existential -pebble game and then modify the original problem by augmenting it with the constraints expressed by the largest winning strategy. This basic result makes it possible to establish deeper connections between pebble games, consistency properties, and tractability of constraint satisfaction. In particular, we use existential -pebble games to introduce the concept of -locality and show that it constitutes a new tractable case of constraint satisfaction that properly extends the well known case in which establishing strong -consistency implies global consistency.