Submodular Constraints and Planar Constraint Networks: New Results

In this paper, we present fast polynomial-time algorithms for solving classes of submodular constraints over Boolean domains. We pose the identified classes of problems within the general framework of Weighted Constraint Satisfaction Problems (WCSPs), reformulated as minimum weighted vertex cover problems. We examine the Constraint Composite Graphs (CCGs) associated with these WCSPs and provide simple arguments for establishing their tractability. We construct simple almost trivial bipartite graph representations for submodular cost functions, and reformulate these WCSPs as max-flow problems on bipartite graphs. By doing this, we achieve better time complexities than state-ofthe-art algorithms. We also use CCGs to exploit planarity in variable interaction graphs, and provide algorithms with significantly improved time complexities for classes of submodular constraints. Moreover, our framework for exploiting planarity is not limited to submodular constraints. Our work confirms the usefulness of studying CCGs associated with combinatorial problems modeled as WCSPs.