Sparse Submodular Function Minimization

In this paper we study the problem of minimizing a submodular function f : 2(V) -> R that is guaranteed to have a k-sparse minimizer. We give a deterministic algorithm that computes an additive epsilon-approximate minimizer of such f in (O) over tilde (poly(k) log(|f|/epsilon) parallel depth using a polynomial number of queries to an evaluation oracle of f, where |f| = max(S subset of V) |f(S)|. Further, we give a randomized algorithm that computes an exact minimizer of f with high probability using (O) over tilde(|V|.poly(k)) queries and polynomial time. When k = (O) over tilde (1), our algorithms use either nearly-constant parallel depth or a nearly-linear number of evaluation oracle queries. All previous algorithms for this problem either use delta(|V|) parallel depth or Omega(|V|(2)) queries. In contrast to state-of-the-art weakly-polynomial and strongly-polynomial time algorithms for SFM, our algorithms use first-order optimization methods, e.g., mirror descent and follow the regularized leader. We introduce what we call sparse dual certificates, which encode information on the structure of sparse minimizers, and both our parallel and sequential algorithms provide new algorithmic tools for allowing first-order optimization methods to efficiently compute them. Correspondingly, our algorithm does not invoke fast matrix multiplication or general linear system solvers and in this sense is more combinatorial than previous state-of-the-art methods.