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 f1, ... , fk on V. The objective is to partition V into k (possibly empty) sets A1, ... , Ak such that the sum Σi=1k fi(Ai) 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 convexprogramming relaxation obtained via the Lovasz-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), HΔ}-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 nodeweighted multiway cut problem in graphs.

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