Beating the logarithmic lower bound: Randomized preemptive disjoint paths and call control algorithms

We consider the maximum disjoint paths problem and its generalization, the call control problem, in the on-line setting. In the maximum disjoint paths problem, we are given a sequence of connection requests for some communication network. Each request consists of a pair of nodes, that wish to communicate over a path in the network. The request has to be immediately connected or rejected, and the goal is to maximize the number of connected pairs, such that no two paths share an edge. In the call control problem, each request has an additional bandwidth specification, and the goal is to maximize the total bandwidth of the connected pairs (throughput), while satisfying the bandwidth constraints (assuming each edge has unit capacity). These classical problems are central in routing and admission control in high speed networks and in optical networks. We present the first known constant-competitive algorithms for both problems on the line. This settles an open problem of Garay et al. and of Leonardi. Moreover, to the best of our knowledge, all previous algorithms for any of these problems, are Ω(log n)-competitive, where n is the number of vertices in the network (and obviously non-competitive for the continuous line). Our algorithms are randomized and preemptive. Our results should be contrasted with the Ω(log n) lower bounds for deterministic preemptive algorithms of Garay et al. and the Ω(log n) lower bounds for randomized non-preemptive algorithms of Lipton and Tomkins and Awerbuch et al. Also, non-constant lower bounds proved by Canetti and Irani for randomized preemptive algorithms for related problems emphasize the distinction between problems that have constant competitive algorithms and other that do not.