Deterministic Algorithms for Submodular Maximization Problems

TL;DR

This paper introduces a derandomization technique for submodular maximization algorithms, showing that deterministic algorithms can match or improve upon randomized ones in certain cases, such as achieving a 1/2-approximation for unconstrained maximization.

Contribution

The authors develop a new derandomization method that eliminates the need for randomness in submodular maximization algorithms, matching or surpassing previous randomized approximation ratios.

Findings

Deterministic algorithms can achieve a 1/2-approximation for unconstrained submodular maximization.

The new technique maintains a small distribution over algorithm states and updates it using linear programming solutions.

Randomization is not necessary for optimal approximation ratios in certain submodular maximization problems.

Abstract

Randomization is a fundamental tool used in many theoretical and practical areas of computer science. We study here the role of randomization in the area of submodular function maximization. In this area most algorithms are randomized, and in almost all cases the approximation ratios obtained by current randomized algorithms are superior to the best results obtained by known deterministic algorithms. Derandomization of algorithms for general submodular function maximization seems hard since the access to the function is done via a value oracle. This makes it hard, for example, to apply standard derandomization techniques such as conditional expectations. Therefore, an interesting fundamental problem in this area is whether randomization is inherently necessary for obtaining good approximation ratios. In this work we give evidence that randomization is not necessary for obtaining good algorithms by presenting a new technique for derandomization of algorithms for submodular function maximization. Our high level idea is to maintain explicitly a (small) distribution over the states of the algorithm, and carefully update it using marginal values obtained from an extreme point solution of a suitable linear formulation. We demonstrate our technique on two recent algorithms for unconstrained submodular maximization and for maximizing submodular function subject to a cardinality constraint. In particular, for unconstrained submodular maximization we obtain an optimal deterministic $1/2$-approximation showing that randomization is unnecessary for obtaining optimal results for this setting.