Localized Client-Server Load Balancing without Global Information

We consider distributed algorithms for maximizing throughput in a network of clients and servers, modeled as a bipartite graph. We seek algorithms and lower bounds for decentralized algorithms in which each participant has only local knowledge about the state of itself and its neighbors. Our problem is analogous to recent work on oblivious routing [M. Bienkowski, M. Korzeniowski, and H. Racke, Proceedings of the $15$th Annual ACM Symposium on Parallel Algorithms and Architectures, 2003, pp. 24-33, C. Harrelson, K. Hildrum, and S. Rao, Proceedings of the $15$th Annual ACM Symposium on Parallel Algorithms and Architectures, 2003, pp. 34-43, H. Racke, Proceedings of the $43$rd Annual IEEE Symposium on Foundations of Computer Science, 2002, pp. 43-52] but with the objective of maximizing throughput rather than minimizing congestion. In contrast to that work, we prove a strong lower bound (polynomial in $n$, the size of the graph) on the competitive ratio of any oblivious algorithm. This is accompanied by simple algorithms achieving upper bounds which are tight in terms of $\OPT$, the maximum throughput achievable by an omniscient algorithm, and are also tight in terms of $m$, the number of servers. Finally, we investigate an online version of the problem, in a restricted model which requires that clients, upon becoming active, must remain so for at least $log(n)$ time steps. In contrast to our primarily negative results in the oblivious case, here we present an algorithm which is constant-competitive. Our lower bounds justify the intuition, implicit in earlier work on the subject [B. Awerbuch and Y. Azar, Proceedings of the $35$th Annual IEEE Symposium on Foundations of Computer Science, 1994, pp. 240-249], that some such restriction (i.e., requiring some stability in the demand pattern over time) is necessary in order to achieve a constant—or even polylogarithmic—competitive ratio.