Bounding the Greedy Strategy in Finite-Horizon String Optimization

TL;DR

This paper establishes new theoretical bounds on the performance of greedy algorithms for finite-horizon string optimization problems, introducing concepts like $K$-submodularity and curvature to tighten these bounds.

Contribution

It introduces $K$-submodularity, $K$-GO-concavity, and curvature-based bounds, providing weaker conditions for greedy strategy performance guarantees in string optimization.

Findings

Greedy strategy bounded by $(1-(1-1/K)^K)$ under new conditions.

Tighter bounds achieved using curvature $ ext{eta}$, with factor $(1/ ext{eta})(1-e^{- ext{eta}})$.

Application examples show improved parameter conditions over previous results.

Abstract

We consider an optimization problem where the decision variable is a string of bounded length. For some time there has been an interest in bounding the performance of the greedy strategy for this problem. Here, we provide weakened sufficient conditions for the greedy strategy to be bounded by a factor of $(1-(1-1/K)^K)$, where $K$ is the optimization horizon length. Specifically, we introduce the notions of $K$-submodularity and $K$-GO-concavity, which together are sufficient for this bound to hold. By introducing a notion of \emph{curvature} $\eta\in(0,1]$, we prove an even tighter bound with the factor $(1/\eta)(1-e^{-\eta})$. Finally, we illustrate the strength of our results by considering two example applications. We show that our results provide weaker conditions on parameter values in these applications than in previous results.