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 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 several 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 ∆ }-approximation for the hypergraph multiway cut problem when ∆ 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.
It is shown in REF that a convex programming relaxation via the Lovász extension follows naturally for MINIMUM SUBMODULAR-COST ALLOCATION, and that θrounding based algorithms provide a unified way to understand and extend several previous results.
Submodular Cost Allocation Problem and Applications
ArXiv
ArXiv
Mathematics Computer Science