Discretely beyond $1/e$: Guided Combinatorial Algortihms for Submodular Maximization

Yixin Chen, Ankur Nath, Chunli Peng, Alan Kuhnle

Keywords: combinatorial algorithms, deterministic algorithms, submodular optimization

TL;DR: First combinatorial algorithms for submodular maximization with ratio better than 1/e ≈ 0.367

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 $0.377$.

Primary Area: Optimization (convex and non-convex, discrete, stochastic, robust)

Submission Number: 18570