Local search heuristic for k-median and facility location problems
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
SIAM Journal on Computing
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
Local Search Heuristics for k-Median and Facility Location Problems
SIAM Journal on Computing
A general approach for incremental approximation and hierarchical clustering
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Oblivious medians via online bidding
LATIN'06 Proceedings of the 7th Latin American conference on Theoretical Informatics
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In the incremental version of the well-known k-medianproblem, the objective is to compute an incremental sequence of facility sets F"1@?F"2@?...@?F"n, where each F"k contains at most k facilities. We say that this incremental medians sequence is R-competitive if the cost of each F"k is at most R times the optimum cost of k facilities. The smallest such R is called the competitive ratio of the sequence {F"k}. Mettu and Plaxton [Ramgopal R. Mettu, C. Greg Plaxton, The online median problem, in: Proc. 41st Symposium on Foundations of Computer Science, FOCS, IEEE, 2000, pp. 339-348; Ramgopal R. Mettu, C. Greg Plaxton, The online median problem, SIAM Journal on Computing 32 (3) (2003) 816-832] presented a polynomial-time algorithm that computes an incremental sequence with competitive ratio ~30. They also showed a lower bound of 2. The upper bound on the ratio was improved to 8 in [Guolong Lin, Chandrashekha Nagarajan, Rajmohan Rajamaran, David P. Williamson, A general approach for incremental approximation and hierarchical clustering, in: Proc. 17th Symposium on Discrete Algorithms, SODA, 2006, pp. 1147-1156] and [Marek Chrobak, Claire Kenyon, John Noga, Neal Young, Online medians via online bidding, in: Proc. 7th Latin American Theoretical Informatics Symposium, LATIN, in: Lecture Notes in Computer Science, vol. 3887, 2006, pp. 311-322]. We improve both bounds in this paper. We first show that no incremental sequence can have competitive ratio better than 2.01 and we give a probabilistic construction of a sequence whose competitive ratio is at most 2+42~7.656. We also propose a new approach to the problem that for instances that we refer to as equable achieves an optimal ratio of 2.