New Approximability Results for the Robust k-Median Problem

Abstract

We consider a robust variant of the classical k-median problem, introduced by Anthony et al. [2]. In the Robust k-Median problem, we are given an n-vertex metric space (V, d) and m client setsThe objective is to open a set F ⊆ V of k facilities such that the worst case connection cost over all client sets is minimized; in other words, minimize max i v∈Si d (F, v). Anthony et al. showed an O(log m) approximation algorithm for any metric and APX-hardness even in the case of uniform metric. In this paper, we show that their algorithm is nearly tight by providing Ω(log m/ log log m) approximation hardness, unless NP ⊆ δ>0 DTIME(2 n δ ). This hardness result holds even for uniform and line metrics. To our knowledge, this is one of the rare cases in which a problem on a line metric is hard to approximate to within logarithmic factor. We complement the hardness result by an experimental evaluation of different heuristics that shows that very simple heuristics achieve good approximations for realistic classes of instances.

Approximating Fair Clustering with Cascaded Norm Objectives

“…In this setting, Theorem 1.5 gives a constant factor approximation which is essentially optimal. This bound implies an O ( log n log log n ) 1/p -approximation for (p, ∞)-Fair Clustering 2 , which matches the recent approximation algorithm of [Makarychev and Vakilian, 2021] and the hardness result of [Bhattacharya et al, 2014]. Thus, for any value of p, the approximation guarantee of Theorem 1.5 smoothly interpolates between the optimal approximation bounds for the previously studied special cases of q = ∞ and q = p.…”

Nearly Optimal Dynamic $k$-Means Clustering for High-Dimensional Data

“…(Bar-Yossef et al, 2002;Feigenbaum et al, 2005;Baswana, 2008;Kelner and Levin, 2011;Ahn et al, 2012b,a;Goel et al, 2012b,a;Ahn et al, 2012a;Baswana et al, 2012;Crouch et al, 2013;Ahn et al, 2013;McGregor, 2014;Baswana et al, 2015;Bhattacharya et al, 2015;Bernstein and Stein, 2016;Abraham et al, 2016a,b;Boutsidis et al, 2016;Kapralov et al, 2017;Song et al, 2017a,b). In addition, k-means and k-median were studied in various different settings, e.g., (Charikar et al, 1998;Indyk and Price, 2011;Backurs et al, 2016;Bhattacharya et al, 2014;Sohler and Woodruff, 2018).…”

Independent Set in k-Claw-Free Graphs: Conditional $$\chi $$-Boundedness and the Power of LP/SDP Relaxations

“…While k-MEANS, k-MEDIAN, and k-CENTER admit constant-factor approximations, it is not very surprising that ROBUST (k, z)-CLUSTERING is harder due to its generality: Makarychev and Vakilian [103] design a polynomial-time O (log m/ log log m)-approximation algorithm, which is tight under a standard complexity assumption [24]. As this precludes the existence of efficient constant-factor approximation algorithms, recent works have focused on designing constant-factor parameterized approximation algorithms.…”