Submodular Maximization with Nearly-optimal Approximation and Adaptivity in Nearly-linear Time

TL;DR

This paper introduces a nearly-linear time algorithm for maximizing monotone submodular functions under a cardinality constraint, achieving near-optimal approximation with significantly reduced adaptivity compared to previous methods.

Contribution

The authors present the first algorithm that attains a $1 - 1/e - ext{epsilon}$ approximation with $O(rac{ ext{log} n}{ ext{epsilon}^2})$ rounds of adaptivity, improving over the previous $ ext{Omega}(n)$ adaptivity bounds.

Findings

Achieves near-optimal approximation with logarithmic adaptivity rounds.

Runs in nearly-linear time with respect to the size of the ground set.

Reduces adaptivity complexity from linear to logarithmic in n.

Abstract

In this paper, we study the tradeoff between the approximation guarantee and adaptivity for the problem of maximizing a monotone submodular function subject to a cardinality constraint. The adaptivity of an algorithm is the number of sequential rounds of queries it makes to the evaluation oracle of the function, where in every round the algorithm is allowed to make polynomially-many parallel queries. Adaptivity is an important consideration in settings where the objective function is estimated using samples and in applications where adaptivity is the main running time bottleneck. Previous algorithms achieving a nearly-optimal $1 - 1/e - \epsilon$ approximation require $\Omega(n)$ rounds of adaptivity. In this work, we give the first algorithm that achieves a $1 - 1/e - \epsilon$ approximation using $O(\ln{n} / \epsilon^2)$ rounds of adaptivity. The number of function evaluations and additional running time of the algorithm are $O(n \mathrm{poly}(\log{n}, 1/\epsilon))$.