Improved Approximation Algorithms for the Min-Max Selecting Items Problem

TL;DR

This paper presents a simple deterministic approximation algorithm for the Min-Max Selecting Items problem with an improved ratio of O(log K / log log K), outperforming previous methods and maintaining polynomial runtime.

Contribution

It introduces a simpler, more efficient approximation algorithm with a better ratio and analyzes the LP formulation's integrality gap for the problem.

Findings

Achieved an O(log K / log log K) approximation ratio.

Proved the LP formulation has an integrality gap of at least Ω(log K / log log K).

Algorithm runs in polynomial time in the RAM model.

Abstract

We give a simple deterministic $O(\log K / \log\log K)$ approximation algorithm for the Min-Max Selecting Items problem, where $K$ is the number of scenarios. While our main goal is simplicity, this result also improves over the previous best approximation ratio of $O(\log K)$ due to Kasperski, Kurpisz, and Zieli\'nski (Information Processing Letters (2013)). Despite using the method of pessimistic estimators, the algorithm has a polynomial runtime also in the RAM model of computation. We also show that the LP formulation for this problem by Kasperski and Zieli\'nski (Annals of Operations Research (2009)), which is the basis for the previous work and ours, has an integrality gap of at least $\Omega(\log K / \log\log K)$.