Scheduling connections via path and edge multicoloring

Abstract

We consider path multicoloring problems, in which one is given a collection of paths defined on a graph and is asked to color some or all of them so as to optimize certain objective functions. Typical objectives are: (a) the minimization of the average, over all edges, of the maximum-multiplicity color when the number of colors is given (MinAvgMult-PMC), (b) the minimization of the number of colors when the maximum multiplicity for each edge is given (Min-PMC), or (c) the maximization of the number of colored paths when both the number of colors and a maximum multiplicity constraint for each edge are given (Max-PMC). Such problems also capture edgemulticoloring variants (such as MinAvgMult-EMC, Min-EMC, and MaxEMC) as special cases and find numerous applications in resource allocation, most notably in optical and wireless networks, and in communication task scheduling. Our contribution is two-fold: On the one hand, we give an exact polynomial-time algorithm for Min-PMC on spider networks with even admissible color multiplicities on each edge. On the other hand, we present an approximation algorithm for MinAvgMult-PMC in star networks, with a ratio strictly better than 2; our algorithm uses an appropriate path orientation. We also show that any algorithm which is based on path orientation cannot achieve an approximation ratio better than 7/6. Our results apply to the corresponding edge multicoloring problems as well.