When planning problems have many kinds of resources or high concurrency, each optimal state has exponentially many minor variants, some of which are "better" than others. Standard methods like \Astar cannot effectively exploit these minor relative differences, and therefore must explore many redundant, clearly suboptimal plans. We describe a new optimal search algorithm for planning that leverages a partial order relation between states. Under suitable conditions, states that are dominated by other states with respect to this order can be pruned while provably maintaining optimality. We also describe a simple method for automatically discovering compatible partial orders in both serial and concurrent domains. In our experiments we find that more than 98% of search states can be pruned in some domains.

This article presents the state-of-the-art in optimal solution methods for decentralized partially observable Markov decision processes (Dec-POMDPs), which are general models for collaborative multiagent planning under uncertainty. Building off the generalized multiagent A* (GMAA*) algorithm, which reduces the problem to a tree of one-shot collaborative Bayesian games (CBGs), we describe several advances that greatly expand the range of Dec-POMDPs that can be solved optimally. First, we introduce lossless incremental clustering of the CBGs solved by GMAA*, which achieves exponential speedups without sacrificing optimality. Second, we introduce incremental expansion of nodes in the GMAA* search tree, which avoids the need to expand all children, the number of which is in the worst case doubly exponential in the node's depth. This is particularly beneficial when little clustering is possible. In addition, we introduce new hybrid heuristic representations that are more compact and thereby enable the solution of larger Dec-POMDPs. We provide theoretical guarantees that, when a suitable heuristic is used, both incremental clustering and incremental expansion yield algorithms that are both complete and search equivalent. Finally, we present extensive empirical results demonstrating that GMAA*-ICE, an algorithm that synthesizes these advances, can optimally solve Dec-POMDPs of unprecedented size.