A polynomial time algorithm for finding a minimum 4-partition of a submodular function

In this paper, we study the minimum k -partition problem of submodular functions, i.e., given a finite set V and a submodular function f:2^V→ℝ , computing a k -partition { V_1, … , V_k } of V with minimum ∑ _i=1^k f(V_i) . The problem is a natural generalization of the minimum k -cut problem in graphs and hypergraphs. It is known that the problem is NP-hard for general k , and solvable in polynomial time for fixed k ≤ 3 . In this paper, we construct the first polynomial-time algorithm for the minimum 4-partition problem.