Multi-Agent and Multivariate 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. We present a generic reduction that transforms a multi-agent (and more generally multivariate) problem into a single-agent one, showing that several properties of the objective function and family of feasible sets are preserved. For maximization, this allows one to leverage algorithmic results from the single-agent submodular setting for the multi-agent case. We are not aware of work for general families in the maximization setting and so these results substantially expand the family of tractable models. For instance, we see that multi-agent (and multivariate) maximization subject to a p-matroid constraint can be reduced to a single-agent problem over the same type of constraint (i.e., p matroid intersections). Allowing the (much) more general class of multivariate submodular objective functions f(S1, S2, . . . , Sk) (instead of decomposable functions f = ∑ i fi) gives new modelling capabilities. We describe a family of multivariate submodular objectives based on quadratic functions and show how these may be used to penalize competition between agents. For monotone multi-agent minimization we give three types of reductions. The first two are black-box reductions which show that (MASO) has an O(α ·min{k, log(n)})approximation when (SO) over F admits an O(α)-approximation over the natural formulation. The third type of reduction is for the special case when every element in the blocking clutter of F has size at most β. Here we show that β-approximations and β ln(n)-approximations are always available for the single-agent and multi-agent problems respectively. Moreover, virtually all known approximations for single-agent monotone submodular minimization are derived in this way.