New performance guarantees for the greedy maximization of submodular set functions

TL;DR

This paper introduces new tight performance guarantees for greedy algorithms in maximizing nondecreasing submodular functions, improving upon classical bounds and providing guarantees based on solution overlap and problem size.

Contribution

It offers novel performance guarantees for greedy maximization of submodular functions, enhancing classical bounds with overlap and domain size considerations.

Findings

Improved performance guarantees over classical bounds.

Guarantees based on overlap between optimal and greedy solutions.

Tight bounds related to problem domain cardinality.

Abstract

We present new tight performance guarantees for the greedy maximization of nondecreasing submodular set functions. Our main result first provides a performance guarantee in terms of the overlap of the optimal and greedy solutions. As a consequence we improve performance guarantees of Nemhauser, Wolsey and Fisher (1978) and Conforti and Cornu\'ejols (1984) for maximization over subsets, which are at least half the size of the problem domain. As a further application, we obtain a new tight performance guarantee in terms of the cardinality of the problem domain.