Efficient Deterministic Algorithms for Maximizing Symmetric Submodular Functions

TL;DR

This paper introduces new efficient deterministic algorithms for maximizing symmetric submodular functions under various constraints, achieving improved approximation ratios and query complexities over previous randomized or less efficient deterministic methods.

Contribution

The paper presents the first deterministic algorithms with improved approximation ratios and query complexities for symmetric submodular maximization under cardinality, matroid, and packing constraints.

Findings

Deterministic algorithm for cardinality constraint attains 0.432 ratio with O(kn) queries.

Deterministic algorithm for matroid constraint attains 1/3−ε ratio with O(kn log ε^{-1}) queries.

New algorithms for packing constraints and knapsack constraints with improved query complexity and approximation ratios.

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 $0.432$. 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 $O(kn)$ queries. Previously, the best deterministic algorithm attains a $0.385-\epsilon$ ratio and uses $O\left(kn (\frac{10}{9\epsilon})^{\frac{20}{9\epsilon}-1}\right)$ queries. For the matroid constraint, we design a deterministic algorithm that attains a $1/3-\epsilon$ ratio and uses $O(kn\log \epsilon^{-1})$ queries. Previously, the best deterministic algorithm can also attain $1/3-\epsilon$ ratio but it uses much larger $O(\epsilon^{-1}n^4)$ queries. For the packing constraints with a large width, we design a deterministic algorithm that attains a $0.432-\epsilon$ ratio and uses $O(n^2)$ 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 $O(n^4)$ queries. Previously, the best deterministic algorithm attains a $0.316-\epsilon$ ratio and uses $\widetilde{O}(n^3)$ queries.