Better Bounds for Incremental Medians

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 [inline-graphic not available: see fulltext] . 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 . 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.