Polynomial Time Preemptive Sum-Multicoloring on Paths

Abstract

The preemptive Sum-Multicoloring (pSMC) problem is a scheduling problem where pairwise conflicting jobs are represented by a conflict graph. The time demands of jobs are given by integer weights on the nodes. The goal is to schedule the jobs in such a way that the sum of their finish times is minimized. We give an ${\mathcal O}(n \cdot {\rm min}(n,{\rm log}\ p))$ time algorithm for pSMC on paths and cycles, where n is the number of nodes and p is the largest time demand. This is the first polynomial algorithm for this problem. It answers the question raised in [8] about the hardness of this problem. In this respect our result identifies a gap between binary-tree conflict graphs – where the question is NP-hard – and paths. Furthermore, our time bound gets very close to that of ${\mathcal O}(n\cdot {\rm log} \ p/{\rm log log} \ p)$ for the non-preemptive SMC on paths [8] . A detailed version of this paper is available at [3].