String Submodular Functions with Curvature Constraints

TL;DR

This paper enhances approximation guarantees for greedy strategies in optimizing string submodular functions by introducing curvature constraints, leading to tighter bounds and broader applicability under string-matroid constraints.

Contribution

It introduces new curvature measures and improves approximation bounds for greedy algorithms in string submodular optimization problems.

Findings

Greedy strategy achieves at least a (1/σ)(1-e^{-σ})-approximation with total backward curvature σ.

Greedy strategy achieves at least a (1-ε)-approximation with total forward curvature ε.

The paper extends analysis to string-matroid constraints and applications.

Abstract

The problem of objectively choosing a string of actions to optimize an objective function that is string submodular has been considered in [1]. There it is shown that the greedy strategy, consisting of a string of actions that only locally maximizes the step-wise gain in the objective function achieves at least a (1-e^{-1})-approximation to the optimal strategy. This paper improves this approximation by introducing additional constraints on curvatures, namely, total backward curvature, total forward curvature, and elemental forward curvature. We show that if the objective function has total backward curvature \sigma, then the greedy strategy achieves at least a \frac{1}{\sigma}(1-e^{-\sigma})-approximation of the optimal strategy. If the objective function has total forward curvature \epsilon, then the greedy strategy achieves at least a (1-\epsilon)-approximation of the optimal strategy. Moreover, we consider a generalization of the diminishing-return property by defining the elemental forward curvature. We also consider the problem of maximizing the objective function subject to general a string-matroid constraint. We investigate an applications of string submodular functions with curvature constraints.