On Chromatic Sums and Distributed Resource Allocation

AbstractThis paper studies an optimization problem that arises in the context of distributed resource allocation: Given a conflict graph that represents the competition of processors over resources, we seek an allocation under which no two jobs with conflicting requirements are executed simultaneously. Our objective is to minimize theaverage response timeof the system. In alternative formulation this is known as theMinimum Color Sum (MCS)problem (E. Kubicka and A. J. Schwenk, 1989. An introduction to chromatic sums,in“Proceedings of the ACM Computer Science Conference,” pp. 39–45.). We show that the algorithm based on finding 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 improved ratios for the classes of bipartite, bounded-degree, and line graphs. The bound generalizes to a 4ρ-approximation of MCS for classes of graphs for which the maximum independent set problem can be approximated within a factor ofρ. On the other hand, we show that ann1−ε-approximation is NP-hard, for someε>0. For some instances of the resource allocation problem, such as theDining Philosophers, an efficient solution requiresedgecoloring of the conflict graph. We introduce theMinimum Edge Color Sum (MECS)problem which is shown to be NP-hard. We show that a 2-approximation to MECS(G) can be obtained distributively usingcompactcoloring withinO(log2n) communication round