Computing lower bounds for minimum sum coloring and optimum cost chromatic partition

Abstract The Minimum Sum Coloring Problem (MSCP) and Optimum Cost Chromatic Partition Problem (OCCP), variants of the well-known Graph Coloring Problem (GCP), find applications in different domains, such as VLSI design, resource allocation, scheduling, and so on. MSCP and OCCP are much harder than GCP and solving them for large graphs is particularly challenging. In the literature, much effort has been spent to develop upper and lower bounds for MSCP. Lower bounds for MSCP are not only interesting in theory but also useful in practice: lower bounds can be used to accelerate solvers for MSCP and evaluate the quality of heuristic results. In this paper, we propose two new theoretical lower bounds for MSCP and OCCP by exploiting structural properties of them. The two lower bounds are based on (relaxed) values of the chromatic number, independence number and bipartite number of the graph. Experiments on standard benchmarks DIMACS and COLOR show that our lower bounds can improve previous known theoretical and computational lower bounds for MSCP on at least 22% of the benchmark instances.