Unconstrained submodular maximization with constant adaptive complexity

Abstract

In this paper, we consider the unconstrained submodular maximization problem. We propose the first algorithm for this problem that achieves a tight (1/2−ε)-approximation guarantee using O(ε −1 ) adaptive rounds and a linear number of function evaluations. No previously known algorithm for this problem achieves an approximation ratio better than 1/3 using less than Ω(n) rounds of adaptivity, where n is the size of the ground set. Moreover, our algorithm easily extends to the maximization of a non-negative continuous DR-submodular function subject to a box constraint, and achieves a tight (1/2 − ε)-approximation guarantee for this problem while keeping the same adaptive and query complexities.

Submodular Maximization with Nearly Optimal Approximation, Adaptivity and Query Complexity

“…Their approaches use multilinear extensions and thus require Ω(nk 2 log 2 (n)) function evaluations to simulate an oracle for ∇f with high enough accuracy. There have also been significant advancements in low-adaptivity algorithms for the problem of unconstrained submodular maximization (Chen et al, 2018;Ene et al, 2018a).…”

“…An essentially optimal algorithm for unconstrained submodular maximization was recently given in (Chen et al, 2018), which allows us to slightly improve the approximation factor of our non-monotone maximization algorithm. Theorem 2.6.…”

“…Theorem 2.6. (Chen et al, 2018) There is an algorithm that achieves a (1/2 − ε)-approximation for unconstrained submodular maximization using O(log(1/ε)/ε) adaptive rounds and O(n log 3 (1/ε)/ε 4 ) evaluation oracle queries.…”

“…The building blocks of our algorithm are the THRESHOLD-SAMPLING subroutine in (Fahrbach et al, 2019), which returns a subset of high-valued elements in O(log(n)/ε) adaptive rounds, and the unconstrained submodular maximization algorithm in (Chen et al, 2018) that gives a (1/2 − ε)approximation in O(log(1/ε)/ε) adaptive rounds. We modify THRESHOLD-SAMPLING to terminate early if its pool of candidate elements becomes too small, which ensures that each element is not chosen with at least constant probability.…”

Scalable Bicriteria Algorithms for Non-Monotone Submodular Cover

“…A number of approximation algorithms have been proposed for unconstrained submodular maximization [Feige et al, 2011;Buchbinder et al, 2015;Dobzinski and Mor, 2015;Buchbinder and Feldman, 2018;Ene et al, 2018;Chen et al, 2019]. One approach is via local search algorithms [Feige et al, 2011;Gharan and Vondrák, 2011;Dobzinski and Mor, 2015], of which the best deterministic approximation ratio is 0.4 [Dobzinski and Mor, 2015] and the best randomized is 0.41 [Gharan and Vondrák, 2011].…”