ON CHROMATIC SUMS AND DISTRIBUTED RESOURCE-ALLOCATION

This paper studies an optimization problem that arises in the context of distributed resource allocation: Given a conflict graph that repres ents the competition of processors over resources, we seek an allocati on under which no two jobs with conflicting requirements are executed simultaneously. Our objective is to minimize the average response time of the system. In alternative formulation this is known as the Minimu m Color Sum (MCS) problem (E. Kubicka and A. J. Schwenk, 1989. An intr oduction to chromatic sums, in ''Proceedings of the ACM Computer Scien ce Conference,'' pp. 39-45.). We show that the algorithm based on find ing iteratively a maximum independent set (MaxIS) is a 4-approximation to the MCS. This bound is tight to within a factor of 2. We give impr oved ratios for the classes of bipartite, bounded-degree, and line gra phs. The bound generalizes to a 4p-approximation of MCS for classes of graphs for which the maximum independent set problem can be approxima ted within a factor of rho. On the other hand, we show that an n(1-eps ilon)-approximation is NP-hard, for some epsilon > 0. For some instanc es of the resource allocation problem, such as the Dining Philosophers , an efficient solution requires edge coloring of the conflict graph. We introduce the Minimum Edge Color Sum (MECS) problem which is shown to be NP-hard. We show that a 2-approximation to MECS(G) can be obtain ed distributively using compact coloring within O(log(2) n) communicat ion rounds. (C) 1998 Academic Press.