Solution Sets in DCOPs and Graphical Games

A distributed constraint optimization problem (DCOP) is a formalism that captures the rewards and costs of local interactions within a team of agents, each of whom is choosing an individual action. When rapidly selecting a single joint action for a team, we typically solve DCOPs (often using locally optimal algorithms) to generate a single solution. However, in scenarios where a set of joint actions (i.e. a set of assignments to a DCOP) is to be generated, metrics are needed to help appropriately select this set and efficiently allocate resources for the joint actions in the set. To address this need, we introduce k-optimality, a metric that captures the desirable properties of diversity and relative quality of a set of locally-optimal solutions using a parameter that can be tuned based on the level of these properties required. To achieve effective resource allocation for this set, we introduce several upper bounds on the cardinalities of k-optimal joint action sets. These bounds are computable in constant time if we ignore the graph structure, but tighter, graphbased bounds are feasible with higher computation cost. Bounds help choose the appropriate level of k-optimality for settings with fixed resources and help determine appropriate resource allocation for settings where a fixed level of k-optimality is desired. In addition, our bounds for a 1-optimal joint action set for a DCOP also apply to the number of pure-strategy Nash equilibria in a graphical game of noncooperative agents.