Min CSP on Four Elements: Moving beyond Submodularity

We report new results on the complexity of the valued constraint satisfaction problem (VCSP). Under the unique games conjecture, the approximability of nite-valued VCSP is fairly well-understood. However, there is yet no characterisation of VCSPs that can be solved exactly in polynomial time. This is unsatisfactory, since such results are interesting from a combinatorial optimisation perspective; there are deep connections with, for instance, submodular and bisubmodular minimisation. We consider the Min and Max CSP problems (i.e. where the cost functions only attain values in {0, 1}) over four-element domains and identify all tractable fragments. Similar classi cations were previously known for twoand three-element domains. In the process, we introduce a new class of tractable VCSPs based on a generalisation of submodularity. We also extend and modify a graph-based technique by Kolmogorov and šivný (originally introduced by Takhanov) for e ciently obtaining hardness results in our setting. This allow us to prove the result without relying on computer-assisted case analyses (which is fairly common when studying VCSPs). The hardness results are further simpli ed by the introduction of powerful reduction techniques.