Colouring weighted bipartite graphs with a co-site constraint

Abstract

We consider a problem of colouring the vertices of a graph, using the integers for colours. For each vertex v there is a given number w v of colours which must be assigned to v. Distinct colours assigned to a given vertex must differ by at least k 0 , and colours assigned to adjacent vertices must differ by at least k 1 . The objective is to minimise the difference between the greatest and smallest colour used for the assignment. This problem is motivated by frequency spectrum management. We introduce fast algorithms for bipartite graphs which find optimal assignments if k 0 is a multiple of k 1 or if k 0⩾3k 1 .