Submodular cost allocation problem and applications

We study the Minimum Submodular-Cost Allocation problem (MSCA). In this problem we are given a finite ground set V and k non-negative submodular set functions f ₁,....,f _k on V. The objective is to partition V into k (possibly empty) sets A ₁,⋯, A _k such that the sum ∑_i=1 ^k f _i (A _i ) is minimized. Several well-studied problems such as the non-metric facility location problem, multiway-cut in graphs and hypergraphs, and uniform metric labeling and its generalizations can be shown to be special cases of MSCA. In this paper we consider a convex-programming relaxation obtained via the Lovász-extension for submodular functions. This allows us to understand several previous relaxations and rounding procedures in a unified fashion and also develop new formulations and approximation algorithms for related problems. In particular, we give a (1.5 - 1/k)-approximation for the hypergraph multiway partition problem. We also give a min {2(1 - 1/k), H _Δ}- approximation for the hypergraph multiway cut problem when Δ is the maximum hyperedge size. Both problems generalize the multiway cut problem in graphs and the hypergraph cut problem is approximation equivalent to the node-weighted multiway cut problem in graphs.