A hypocoloring model for batch scheduling

Starting from a batch scheduling problem, we consider a weighted subcoloring in a graph G; each node v has a weight w(v); each color class S is a subset of nodes which generates a collection of node disjoint cliques. The weight w(S) is defined as max{w(K)=∑v∈Kw(v)|K∈S}. In the scheduling problem, the completion time is given by ∑i=1kw(Si) where S=(S1,…,Sk) is a partition of the node set of graph G into color classes as defined above. Properties of such colorings concerning special classes of graphs (line graphs of cacti, block graphs) are stated; complexity and approximability results are presented. The associated decision problem is shown to be NP-complete for bipartite graphs with maximum degree at most 39 and triangle-free planar graphs with maximum degree k for any k⩾3. Polynomial algorithms are given for graphs with maximum degree two and for the forests with maximum degree k. An (exponential) algorithm based on a simple separation principle is sketched for graphs without triangles.