eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-07-04
29:1
29:14
10.4230/LIPIcs.ICALP.2018.29
article
Interpolating between k-Median and k-Center: Approximation Algorithms for Ordered k-Median
Chakrabarty, Deeparnab
1
Swamy, Chaitanya
2
Dept. of Computer Science, Dartmouth College, Hanover, NH 03755-3510, USA
Dept. of Combinatorics and Optimization, Univ. Waterloo, Waterloo, ON N2L 3G1, Canada
We consider a generalization of k-median and k-center, called the ordered k-median problem. In this problem, we are given a metric space (D,{c_{ij}}) with n=|D| points, and a non-increasing weight vector w in R_+^n, and the goal is to open k centers and assign each point j in D to a center so as to minimize w_1 *(largest assignment cost)+w_2 *(second-largest assignment cost)+...+w_n *(n-th largest assignment cost). We give an (18+epsilon)-approximation algorithm for this problem. Our algorithms utilize Lagrangian relaxation and the primal-dual schema, combined with an enumeration procedure of Aouad and Segev. For the special case of {0,1}-weights, which models the problem of minimizing the l largest assignment costs that is interesting in and of by itself, we provide a novel reduction to the (standard) k-median problem, showing that LP-relative guarantees for k-median translate to guarantees for the ordered k-median problem; this yields a nice and clean (8.5+epsilon)-approximation algorithm for {0,1} weights.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol107-icalp2018/LIPIcs.ICALP.2018.29/LIPIcs.ICALP.2018.29.pdf
Approximation algorithms
Clustering
Facility location
Primal-dual method