On Chromatic Sums and Distributed Resource Allocation

Amotz Bar-Noy
Department of Electrical Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel.

Mihir Bellare Department of Computer Science and Engineering, University of California at San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA.

Magnús M. Halldórsson
Science Institute, University of Iceland, IS-107 Reykjavik, Iceland.

Hadas Shachnai
The Department of Computer Science, Technion, Haifa 32000, Israel.

Tami Tamir
Israel Software Lab, Intel, Haifa, Israel.

This 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 the average response time of the system. In alternative formulation this is known as the Minimum Color Sum (MCS) problem. 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 4r-approximation of MCS for classes of graphs for which the maximum independent set problem can be approximated within a factor of r. On the other hand, we show that an n^{1- epsilon}-approximation is NP-hard, for some epsilon > 0. For some instances 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 obtained distributively using compact coloring within O(log^2 n) communication rounds.