Abstract:
In this paper, 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 that were originally introduced in the context of Datalog. The crucial insight that relates pebble games to constraint satisfaction is that the key concept of strong k-consistency is equivalent to a property of winning strategies for a player called the Duplicator in the, so-called, existential k-pebble game. Furthermore, we show that strong k-consistency can be established if and only if the Duplicator wins the existential k pebble game. Moreover, whenever strong k-consistency can be established, one method for doing this is to first compute the largest winning strategy for the Duplicator in the existential k-pebble game and then modify the original problem by augmenting it with the constraints expressed by the largest winning strategy. We use this basic result to derive deeper connections between pebble games, consistency properties, and tractability of constraint satisfaction. Finally, using k-pebble games, we introduce the concept of $k$-locality and show that it constitutes a new tractable case of constraint satisfaction that properly extends the well known case in which establishing strong k-consistency implies global consistency.