Improved approximation algorithms for the Min-Max Selecting Items problem

We give a simple deterministic O(logK/loglogK) 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(logK) due to Kasperski, Kurpisz, and Zieliński (2013) [4]. 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 Zieliński (2009) [6], which is the basis for the previous work and ours, has an integrality gap of at least Ω(logK/loglogK).