Better bounds for incremental medians

In the incremental version of the well-known , the objective is to compute an incremental sequence of facility sets F1⊆F2⊆⋯⊆Fn, where each Fk contains at most k facilities. We say that this incremental medians sequence is R-competitive if the cost of each Fk is at most R times the optimum cost of k facilities. The smallest such R is called the competitive ratio of the sequence {Fk}. 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 . We also propose a new approach to the problem that for instances that we refer to as equable achieves an optimal ratio of 2.