Submodular Maximization with Nearly Optimal Approximation, Adaptivity and Query Complexity

TL;DR

This paper introduces a distributed algorithm for monotone submodular maximization under a cardinality constraint that achieves near-optimal approximation, adaptivity, and query complexity, significantly improving efficiency for large-scale applications.

Contribution

It presents a nearly optimal distributed algorithm with optimal approximation, low adaptivity, and reduced query complexity for submodular maximization and cover problems.

Findings

Achieves a (1-1/e-ε)-approximation in expectation.

Runs in O(log(n)) adaptive rounds with O(n) function evaluations.

Extends results to submodular cover problem.

Abstract

Submodular optimization generalizes many classic problems in combinatorial optimization and has recently found a wide range of applications in machine learning (e.g., feature engineering and active learning). For many large-scale optimization problems, we are often concerned with the adaptivity complexity of an algorithm, which quantifies the number of sequential rounds where polynomially-many independent function evaluations can be executed in parallel. While low adaptivity is ideal, it is not sufficient for a distributed algorithm to be efficient, since in many practical applications of submodular optimization the number of function evaluations becomes prohibitively expensive. Motivated by these applications, we study the adaptivity and query complexity of adaptive submodular optimization. Our main result is a distributed algorithm for maximizing a monotone submodular function with cardinality constraint $k$ that achieves a $(1-1/e-\varepsilon)$-approximation in expectation. This algorithm runs in $O(\log(n))$ adaptive rounds and makes $O(n)$ calls to the function evaluation oracle in expectation. The approximation guarantee and query complexity are optimal, and the adaptivity is nearly optimal. Moreover, the number of queries is substantially less than in previous works. Last, we extend our results to the submodular cover problem to demonstrate the generality of our algorithm and techniques.