Interval Scheduling and Colorful Independent Sets

The np-hard independent set problem is to determine for a given graph g and an integer k whether g contains a set of k pairwise non-adjacent vertices. The problem has numerous applications in scheduling, including resource allocation and steel manufacturing. There, one encounters restricted graph classes such as 2-union graphs, which are edge-wise unions of two interval graphs on the same vertex set, or strip graphs, where additionally one of the two interval graphs is a disjoint union of cliques.we prove np-hardness of independent set on a very restricted subclass of 2-union graphs and identify natural parameterizations to chart the possibilities and limitations of effective polynomial-time preprocessing (kernelization) and fixed-parameter algorithms. Our algorithms benefit from novel formulations of the computational problems in terms of (list-)colored interval graphs.