TY - GEN
T1 - An Improved Local Search Algorithm for k-Median
AU - Cohen-Addad, Vincent
AU - Gupta, Anupam
AU - Oh, Lunjia Hu Hoon
AU - Saulpic, David
N1 - Publisher Copyright:
Copyright © 2022 by SIAM Unauthorized reproduction of this article is prohibited.
PY - 2022
Y1 - 2022
N2 - We present a new local-search algorithm for the k-median clustering problem. We show that local optima for this algorithm give a (2.836 + ϵ)-approximation; our result improves upon the (3 + ϵ)-approximate localsearch algorithm of Arya et al. [AGK+01]. Moreover, a computer-aided analysis of a natural extension suggests that this approach may lead to an improvement over the best-known approximation guarantee for the problem. The new ingredient in our algorithm is the use of a potential function based on both the closest and second-closest facilities to each client. Specifically, the potential is the sum over all clients, of the distance of the client to its closest facility, plus (a small constant times) the truncated distance to its second-closest facility. We move from one solution to another only if the latter can be obtained by swapping a constant number of facilities, and has a smaller potential than the former. This refined potential allows us to avoid the bad local optima given by Arya et al. for the local-search algorithm based only on the cost of the solution.
AB - We present a new local-search algorithm for the k-median clustering problem. We show that local optima for this algorithm give a (2.836 + ϵ)-approximation; our result improves upon the (3 + ϵ)-approximate localsearch algorithm of Arya et al. [AGK+01]. Moreover, a computer-aided analysis of a natural extension suggests that this approach may lead to an improvement over the best-known approximation guarantee for the problem. The new ingredient in our algorithm is the use of a potential function based on both the closest and second-closest facilities to each client. Specifically, the potential is the sum over all clients, of the distance of the client to its closest facility, plus (a small constant times) the truncated distance to its second-closest facility. We move from one solution to another only if the latter can be obtained by swapping a constant number of facilities, and has a smaller potential than the former. This refined potential allows us to avoid the bad local optima given by Arya et al. for the local-search algorithm based only on the cost of the solution.
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M3 - Conference contribution
AN - SCOPUS:85128213546
T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
SP - 1556
EP - 1612
BT - ACM-SIAM Symposium on Discrete Algorithms, SODA 2022
PB - Association for Computing Machinery
T2 - 33rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2022
Y2 - 9 January 2022 through 12 January 2022
ER -