Discretely Beyond $1/e$: Guided Combinatorial Algorithms for Submodular Maximization

TL;DR

This paper introduces new combinatorial algorithms that surpass the $1/e$ approximation barrier for constrained submodular maximization, achieving higher ratios with efficient query complexity.

Contribution

It presents the first combinatorial algorithms to beat the $1/e$ barrier for submodular maximization under size and matroid constraints, using guided greedy and local search techniques.

Findings

Achieved a 0.385 approximation ratio for size constraints.

Achieved a 0.305 approximation ratio for matroid constraints.

Developed deterministic algorithms with the same ratios and complexity.

Abstract

For constrained, not necessarily monotone submodular maximization, all known approximation algorithms with ratio greater than $1/e$ require continuous ideas, such as queries to the multilinear extension of a submodular function and its gradient, which are typically expensive to simulate with the original set function. For combinatorial algorithms, the best known approximation ratios for both size and matroid constraint are obtained by a simple randomized greedy algorithm of Buchbinder et al. [9]: $1/e \approx 0.367$ for size constraint and $0.281$ for the matroid constraint in $\mathcal O (kn)$ queries, where $k$ is the rank of the matroid. In this work, we develop the first combinatorial algorithms to break the $1/e$ barrier: we obtain approximation ratio of $0.385$ in $\mathcal O (kn)$ queries to the submodular set function for size constraint, and $0.305$ for a general matroid constraint. These are achieved by guiding the randomized greedy algorithm with a fast local search algorithm. Further, we develop deterministic versions of these algorithms, maintaining the same ratio and asymptotic time complexity. Finally, we develop a deterministic, nearly linear time algorithm with ratio $0.377$.