Incremental Medians via Online Bidding

In the k-median problem we are given sets of facilities and customers, and distances between them. For a given set F of facilities, the cost of serving a customer u is the minimum distance between u and a facility in F. The goal is to find a set F of k facilities that minimizes the sum, over all customers, of their service costs. Following the work of Mettu and Plaxton, we study the incremental medians problem, where k is not known in advance. An incremental algorithm produces a nested sequence of facility sets F ₁⊆F ₂⊆⋅⋅⋅⊆F _n , where |F _k |=k for each k. Such an algorithm is called c -cost-competitive if the cost of each F _k is at most c times the optimum k-median cost. We give improved incremental algorithms for the metric version of this problem: an 8-cost-competitive deterministic algorithm, a 2e≈5.44-cost-competitive randomized algorithm, a (24+ε)-cost-competitive, polynomial-time deterministic algorithm, and a 6e+ε≈16.31-cost-competitive, polynomial-time randomized algorithm. We also consider the competitive ratio with respect to size. An algorithm is s -size-competitive if the cost of each F _k is at most the minimum cost of any set of k facilities, while the size of F _k is at most sk. We show that the optimal size-competitive ratios for this problem, in the deterministic and randomized cases, are 4 and e. For polynomial-time algorithms, we present the first polynomial-time O(log m)-size-approximation algorithm for the offline problem, as well as a polynomial-time O(log m)-size-competitive algorithm for the incremental problem. Our upper bound proofs reduce the incremental medians problem to the following online bidding problem: faced with some unknown threshold T∈ℝ⁺, an algorithm must submit “bids” b∈ℝ⁺ until it submits a bid b≥T, paying the sum of all its bids. We present folklore algorithms for online bidding and prove that they are optimally competitive. We extend some of the above results for incremental medians to approximately metric distance functions and to incremental fractional medians. Finally, we consider a restricted version of the incremental medians problem where k is restricted to one of two given values, for which we give a deterministic algorithm with a nearly optimal cost-competitive ratio. The conference version of this paper appeared in (Chrobak, M., et al. in Lecture Notes in Computer Science, vol. 3887, pp. 311–322, 2006). Research of M. Chrobak supported by NSF Grant CCR-0208856.