Better Bounds for Incremental Medians

Abstract

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 . Mettu and Plaxton [6,7] 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 [5] and [4]. 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 $2+4\sqrt{2} \approx 7.656$. We also propose a new approach to the problem that for instances that we refer to as equable achieves an optimal competitive ratio of 2.