a b s t r a c t Let (L; ⊓, ⊔) be a finite lattice and let n be a positive integer. A function f :In this article we study submodular functions when L is a diamond. Given oracle access to f we are interested in finding x ∈ L n such that f (x) = min y∈L n f (y) as efficiently as possible. We establish • a min-max theorem, which states that the minimum of the submodular function is equal to the maximum of a certain function defined over a certain polyhedron; and• a good characterisation of the minimisation problem, i.e., we show that given an oracle for computing a submodular f : L n → Z and an integer m such that min x∈L n f (x) = m, there is a proof of this fact which can be verified in time polynomial in n and max t∈L n log |f (t)|; and • a pseudopolynomial-time algorithm for the minimisation problem, i.e., given an oracle for computing a submodular f : L n → Z one can find min t∈L n f (t) in time bounded by a polynomial in n and max t∈L n |f (t)|.

We report new results on the complexity of the valued constraint satisfaction problem (VCSP). Under the unique games conjecture, the approximability of finite-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 classifications were previously known for two-and 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 Živný (originally introduced by Takhanov) for efficiently obtaining hardness results in our setting. This allow us to prove the result without relying on computer-assisted case analyses (which otherwise are fairly common when studying the complexity and approximability of VCSPs.) The hardness results are further simplified by the introduction of powerful reduction techniques.

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