Tight Analysis of a Multiple-Swap Heuristic for Budgeted Red-Blue Median

Budgeted Red-Blue Median is a generalization of classic k-Median in that there are two sets of facilities, say R and B, that can be used to serve clients located in some metric space. The goal is to open kr facilities in R and kb facilities in B for some given bounds kr, kb and connect each client to their nearest open facility in a way that minimizes the total connection cost. We extend work by Hajiaghayi, Khandekar, and Kortsarz [2012] and show that a multiple-swap local search heuristic can be used to obtain a (5 + )-approximation for Budgeted Red-Blue Median for any constant > 0. This is an improvement over their single swap analysis and beats the previous best approximation guarantee of 8 by Swamy [2014]. We also present a matching lower bound showing that for every p ≥ 1, there are instances of Budgeted Red-Blue Median with local optimum solutions for the p-swap heuristic whose cost is 5 + Ω ( 1 p ) times the optimum solution cost. Thus, our analysis is tight up to the lower order terms. In particular, for any > 0 we show the single-swap heuristic admits local optima whose cost can be as bad as 7− times the optimum solution cost.