Efficient deterministic algorithms for maximizing symmetric submodular functions

Abstract

Symmetric submodular maximization is an important class of combinatorial optimization problems, including MAX-CUT on graphs and hyper-graphs. The state-of-the-art algorithm for the problem over general constraints has an approximation ratio of 0.432 [16]. The algorithm applies the canonical continuous greedy technique that involves a sampling process. It, therefore, suffers from high query complexity and is inherently randomized. In this paper, we present several efficient deterministic algorithms for maximizing a symmetric submodular function under various constraints. Specifically, for the cardinality constraint, we design a deterministic algorithm that attains a 0.432 ratio and uses $O\left(kn\right)$ queries. Previously, the best deterministic algorithm attains a $0.385-ϵ$ ratio and uses $O\left(kn{\left(\frac{10}{9ϵ}\right)}^{\frac{20}{9ϵ}-1}\right)$ queries [12]. For the matroid constraint, we design a deterministic algorithm that attains a $1/3-ϵ$ ratio and uses $O(kn\log {ϵ}^{-1})$ queries. Previously, the best deterministic algorithm can also attain $1/3-ϵ$ ratio but it uses much larger $O\left({ϵ}^{-1}{n}^4\right)$ queries [24]. For the packing constraints with a large width, we design a deterministic algorithm that attains a $0.432-ϵ$ ratio and uses $O\left(n^2\right)$ queries. To the best of our knowledge, there is no deterministic algorithm for the constraint previously. The last algorithm can be adapted to attain a 0.432 ratio for single knapsack constraint using $O\left(n^4\right)$ queries. Previously, the best deterministic algorithm attains a $0.316-ϵ$ ratio and uses $\tilde{O}\left(n^3\right)$ queries [2].