Subquadratic Submodular Maximization with a General Matroid Constraint

TL;DR

This work delivers a near-optimal approximation for monotone submodular maximization under a general matroid constraint with subquadratic query complexity. By marrying the continuous-relaxation framework with a novel fast rounding routine based on directed-cycle exchanges, the authors reduce independence-query costs from quadratic to roughly $O(r^{3/2})$ per base-merge, achieving $O(\tilde{O}_{\varepsilon}(\sqrt{r}\,n))$ total queries in many regimes and extending to rank- and dynamic-oracle models. The key technical advance is the directed-cycle-based rounding, which extends prior two-cycle methods to cycles of arbitrary length and uses sampling plus binary-search to locate cycles efficiently. The results substantially improve the practicality of near-optimal submodular maximization under general matroids, with potential impact on large-scale optimization tasks in machine learning and economics where fast oracle-based computation is essential.

Abstract

We consider fast algorithms for monotone submodular maximization with a general matroid constraint. We present a randomized $(1 - 1/e - ε)$-approximation algorithm that requires $\tilde{O}_ε(\sqrt{r} n)$ independence oracle and value oracle queries, where $n$ is the number of elements in the matroid and $r \leq n$ is the rank of the matroid. This improves upon the previously best algorithm by Buchbinder-Feldman-Schwartz [Mathematics of Operations Research 2017] that requires $\tilde{O}_ε(r^2 + \sqrt{r}n)$ queries. Our algorithm is based on continuous relaxation, as with other submodular maximization algorithms in the literature. To achieve subquadratic query complexity, we develop a new rounding algorithm, which is our main technical contribution. The rounding algorithm takes as input a point represented as a convex combination of $t$ bases of a matroid and rounds it to an integral solution. Our rounding algorithm requires $\tilde{O}(r^{3/2} t)$ independence oracle queries, while the previously best rounding algorithm by Chekuri-Vondrák-Zenklusen [FOCS 2010] requires $O(r^2 t)$ independence oracle queries. A key idea in our rounding algorithm is to use a directed cycle of arbitrary length in an auxiliary graph, while the algorithm of Chekuri-Vondrák-Zenklusen focused on directed cycles of length two.

Theorems & Definitions (28)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1: nguyen2019notechakrabarty2019faster
• Theorem 3.1: follows from buchbinder2017comparing and buchbinder2017comparing
• Remark 3.2
• proof : Proof of Theorem \ref{['main_submodular_max']}
• Theorem 5.1
• Definition 5.2
• Proposition 5.3