neal young / Chrobak08Online

  • publication/Chrobak08Online.png Following Mettu and Plaxton, we study incremental algorithms for the \(k\)-medians problem. Such an algorithm must produce a nested sequence \(F_1 \subseteq F_2 \subseteq \cdots \subseteq F_n\) of sets of facilities. Mettu and Plaxton show that incremental metric medians has a (roughly) \(40\)-competitive deterministic polynomial-time algorithm. We give improved algorithms, including a \((24+\epsilon)\)-competitive deterministic polynomial-time algorithm and a \(5.44\)-competitive, randomized, non-polynomial-time algorithm.

    We also 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 \(s k\). We present optimally competitive algorithms for this problem.

    Our proofs reduce incremental medians to the following online bidding problem: faced with some unknown threshold \(T>0\), an algorithm must submit ``bids'' \(b>0\) until it submits a bid as large as \(T\). The algorithm pays the sum of its bids. We describe optimally competitive algorithms for online bidding.

    Our results on cost-competitive incremental medians extend to approximately metric distance functions, incremental fractional medians, and incremental bicriteria approximation.
    Journal version of [2006].

© Copyrights are reserved by the publishers.
Download for personal and limited academic use only.