Multi-Agent Submodular Optimization

Recent years have seen many algorithmic advances in the area of submodular optimization: (SO) min /max f(S) : S ∈ F , where F is a given family of feasible sets over a ground set V and f : 2 → R is submodular. This progress has been coupled with a wealth of new applications for these models. Our focus is on a more general class of multi-agent submodular optimization (MASO) which was introduced by Goel et al. in the minimization setting: min ∑ i fi(Si) : S1 ⊎ S2 ⊎ · · · ⊎ Sk ∈ F . Here we use ⊎ to denote disjoint union and hence this model is attractive where resources are being allocated across k agents, each with its own submodular cost function fi(). In this paper we explore the extent to which the approximability of the multi-agent problems are linked to their single-agent primitives, referred to informally as the multi-agent gap. We present different reductions that transform a multi-agent problem into a single-agent one. For maximization we show that (MASO) admits an O(α)-approximation whenever (SO) admits an α-approximation over the multilinear formulation, and thus substantially expanding the family of tractable models. We also discuss several family classes (such as spanning trees, matroids, and p-systems) that have a provable multi-agent gap of 1. In the minimization setting we show that (MASO) has an O(α · min{k, log(n) log( n logn )})approximation whenever (SO) admits an α-approximation over the convex formulation. In addition, we discuss the class of “bounded blocker” families where there is a provably tight O(logn) gap between (MASO) and (SO).