Oblivious medians via online bidding

Abstract

Following Mettu and Plaxton [22, 21], we study oblivious algorithms for the k-medians problem. Such an algorithm produces an incremental sequence of facility sets. We give improved algorithms, including a (24 + ε)-competitive deterministic polynomial algorithm and a 2e ≈ 5.44-competitive randomized non-polynomial algorithm. Our approach is similar to that of [18], which was done independently. We then consider the competitive ratio with respect to size. An algorithm is s-size-competitive if, for each k, the cost of 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 present optimally competitive algorithms for this problem. Our proofs reduce oblivious medians 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 its bids. We describe optimally competitive algorithms for online bidding. Some of these results extend to approximately metric distance functions, oblivious fractional medians, and oblivious bicriteria approximation. When the number of medians takes only two possible values k or l, for k