Better bounds for incremental medians

Marek Chrobak, Mathilde Hurand
2011 Theoretical Computer Science  
In the incremental version of the well-known k-median problem, 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 }. 816-832] presented a polynomial-time algorithm that computes an incremental
more » ... ence 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: 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 + 4 √ 2 ≈ 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.
doi:10.1016/j.tcs.2009.07.006 fatcat:exhgsfotgnhx3owvmo6opk3coy