Local Search Algorithms for the Red-Blue Median Problem

• Published in: Algorithmica Vol. 63; no. 4; pp. 795 - 814
• Main Authors: Hajiaghayi, M., Khandekar, R., Kortsarz, G.
• Format: Journal Article Conference Proceeding
• Language: English
• Published: New York Springer-Verlag 01-08-2012; Springer
• Subjects: Algorithm Analysis and Problem Complexity; Algorithmics. Computability. Computer arithmetics; Algorithms; Applied sciences; Computer Science; Computer science; control theory; systems; Computer Systems Organization and Communication Networks; Data Structures and Information Theory; Exact sciences and technology; Logistics; Mathematics of Computing; Operational research and scientific management; Operational research. Management science; Theoretical computing; Theory of Computation; Facility location; Prize-collecting; median; Local search algorithms; Local search; Costs; Metric space; Graph theory; Computational complexity; Location problem; Search algorithm; Polynomial time; K median problem; k-median; Best approximation
• ISSN: 0178-4617; 1432-0541
• DOI: 10.1007/s00453-011-9547-9

In this paper, we consider the following
red-blue median
problem which is a generalization of the well-studied
k-median
problem. The input consists of a set of
red
facilities, a set of
blue
facilities, and a set of clients in a metric space and two integers
k
r
,
k
b
≥0. The problem is to open at most
k
r
red facilities and at most
k
b
blue facilities and minimize the sum of distances of clients to their respective closest open facilities.
We show, somewhat surprisingly, that the following simple local search algorithm yields a constant factor approximation for this problem. Start by opening any
k
r
red and
k
b
blue facilities. While possible, decrease the cost of the solution by closing a pair of red and blue facilities and opening a pair of red and blue facilities.
We also improve the approximation factor for the
prize-collecting
k-median
problem from 4 (Charikar et al. in Proceedings of the ACM-SIAM Symposium on Discrete Algorithms, pp. 642–641,
2001
) to 3+
ϵ
, which matches the current best approximation factor for the
k
-median problem.