Sparse Submodular Function Minimization
Andrei Graur, Haotian Jiang, Aaron Sidford
TL;DR
This work addresses submodular function minimization (SFM) under the crucial sparsity assumption that the minimizer is $k$-sparse. It departs from traditional cutting-plane approaches and instead develops a framework based on sparse dual certificates, an extension $f^{\sharp \mathcal{R}}$ tied to a ring family, and arc constraints, enabling first-order optimization methods to drive progress. It delivers two main achievements: (i) a parallel algorithm with nearly-constant depth for fixed $k$ and $\widetilde{O}(|V|\cdot\mathrm{poly}(k))$ query complexity, and (ii) randomized sequential algorithms with near-linear or strongly polynomial EO-query complexity to obtain either an $\varepsilon$-approximate minimizer or an exact minimizer whp. These results improve over prior $\Omega(|V|)$ depth and $\Omega(|V|^2)$ query bounds, and illustrate how sparse dual certificates can integrate first-order optimization with combinatorial structure to yield efficient SFM in sparse regimes, with potential broader applicability beyond this problem.
Abstract
In this paper we study the problem of minimizing a submodular function $f : 2^V \rightarrow \mathbb{R}$ that is guaranteed to have a $k$-sparse minimizer. We give a deterministic algorithm that computes an additive $ε$-approximate minimizer of such $f$ in $\widetilde{O}(\mathsf{poly}(k) \log(|f|/ε))$ parallel depth using a polynomial number of queries to an evaluation oracle of $f$, where $|f| = \max_{S \subseteq V} |f(S)|$. Further, we give a randomized algorithm that computes an exact minimizer of $f$ with high probability using $\widetilde{O}(|V| \cdot \mathsf{poly}(k))$ queries and polynomial time. When $k = \widetilde{O}(1)$, 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 $Ω(|V|)$ parallel depth or $Ω(|V|^2)$ 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 {\em 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.