On the Complexity of Submodular Function Minimisation on Diamonds

Let (L;⊓,⊔) be a finite lattice and let n be a positive integer. A function f : L → R is said to be submodular if f(a ⊓ b) + f(a ⊔ b) ≤ f(a)+f(b) for all a, b ∈ L. In this paper we study submodular functions when L is a diamond. Given oracle access to f we are interested in finding x ∈ L such that f(x) = miny∈Ln 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 → Z and an integer m such that minx∈Ln f(x) = m, there is a proof of this fact which can be verified in time polynomial in n and maxt∈Ln log |f(t)|; and • a pseudo-polynomial time algorithm for the minimisation problem, i.e., given an oracle for computing a submodular f : L → Z one can find mint∈Ln f(t) in time bounded by a polynomial in n and maxt∈Ln |f(t)|.