Better polynomial algorithms for scheduling unit-length jobswith bipartite incompatibility graphs on uniform machines

The goal of this paper is to explore and to provide tools for the investigation of the problems of unit-length scheduling of incompatible jobs on uniform machines. We present two new algorithms that are a significant improvement over the known algorithms. The first one is Algorithm 2 which is 2-approximate for the problem Qm|p_j = 1, G = bisubquartic|C_max. The second one is Algorithm 3 which is 4-approximate for the problem Qm|p_j = 1, G = bisubquartic|ΣC_j, where m ϵ {2, 3, 4}. The theory behind the proposed algorithms is based on the properties of 2-coloring with maximal coloring width, and on the properties of ideal machine, an abstract machine that we introduce in this paper.