Hardness of submodular cost allocation: Lattice matching and a simplex coloring conjecture

Abstract

We consider the Minimum Submodular Cost Allocation (MSCA) problem [3]. In this problem, we are given κ submodular cost functions f1, . . . , fκ : 2V → ℝ+ and the goal is to partition V into κ sets A1, . . . ,Ak so as to minimize the total cost ∑ikappa;=1 fi(Ai). We show that MSCA is inapproximable within any multiplicative factor even in very restricted settings; prior to our work, only Set Cover hardness was known. In light of this negative result, we turn our attention to special cases of the problem. We consider the setting in which each function fi satisfies fi = gi+h, where each gi is monotone submodular and h is (possibly non-monotone) submodular. We give an O(κ log |V |) approximation for this problem. We provide some evidence that a factor of κ may be necessary, even in the special case of Hypergraph Labeling [3]. In particular, we formulate a simplex-coloring conjecture that implies a Unique-Games-hardness of κ-1-∈ for κ-uniform Hypergraph Labeling and label set [κ]. We provide a proof of the simplex-coloring conjecture for κ = 3.1.