Resource-sharing system scheduling and circular chromatic number

A graph G is used as a model for a resource sharing system, where each vertex represents a process and an edge joining two vertices means that the corresponding processes share a resource. A scheduling of G is a mapping f : {1, 2, 3,...} -> 2(V(G)), where f (i) consists of processes that are operating at round i. The rate of f is defined as rate(f) = lim sup(n ->infinity) Sigma(i=1)(n) vertical bar f(i)vertical bar/n vertical bar V (G)vertical bar, which is the average fraction of operating processes at each round. A scheduling is fair if adjacent vertices alternate their turns in operating. The operating rate gamma* (G) of G is the maximum rate of a fair scheduling. Fair schedulings of a graph was first studied by Barbosa and Gafni. They introduced the method of "scheduling by edge reversal" which derives a fair scheduling through an acyclic orientation. Through scheduling by edge reversal, Barbosa and Gafni related gamma* (G) to the structure of acyclic orientations of G. We point out that this relation implies that gamma*(G) is equal to the reciprocal of the circular chromatic number of G. Both circular coloring and scheduling by edge reversal have been studied extensively in the past decade. The former by graph theorists, and the latter by computer scientists. However, it seems that neither side knew the existence of the other. This paper intends to build a connection between the two sides. We show that certain open problems concerning scheduling by edge reversal are indeed solved under the language of circular coloring. In the study of fair scheduling, Barbosa and Gafni defined a variation of multiple coloring of graphs: the interleaved p-color, q-tuple colorings. We formulate the interleaved coloring as a graph homomorphism problem. In the study of circular chromatic number, Bondy and Hell defined (p, q)-colorings and also formulated it as a graph homomorphism problem. We prove that the target graph for the interleaved p-color, q-tuple coloring and the target graph of (p, q)-coloring are homomorphically equivalent. This gives another proof of the fact that gamma*(G) = 1/chi(C)(G). Moreover, the proof gives an explicit formula which deduces an optimal circular coloring of G from an optimal interleaved coloring of G, and vice versa. This paper also introduces two other schedulings of a graph, the weakly fair scheduling and the strongly fair scheduling. It is proved that the rate of an optimal strongly fair scheduling of a graph G is also equal to the reciprocal of the circular chromatic number of G, and the rate of an optimal weakly fair scheduling of G is equal to the reciprocal of the fractional chromatic number of G. Barbosa and Gafni presented an algorithm that determines the rate gamma(o) (w) of the scheduling induced by an acyclic orientation w of G. By using Karp's minimum mean cycle algorithm, we give a faster algorithm to calculate gamma(o)(w). (c) 2005 Elsevier B.V. All rights reserved.