Interval scheduling and colorful independent sets

Abstract

Numerous applications in scheduling, such as resource allocation or steel manufacturing, can be modeled using the NP-hard Independent Set problem (given an undirected graph and an integer $k$, find a set of at least $k$ pairwise non-adjacent vertices). Here, one encounters special graph classes like 2-union graphs (edge-wise unions of two interval graphs) and strip graphs (edge-wise unions of an interval graph and a cluster graph), on which Independent Set remains $\mathrm{NP}$-hard but admits constant ratio approximations in polynomial time. We study the parameterized complexity of Independent Set on 2-union graphs and on subclasses like strip graphs. Our investigations significantly benefit from a new structural “compactness” parameter of interval graphs and novel problem formulations using vertex-colored interval graphs. Our main contributions are as follows:

• 1. We show a complexity dichotomy: restricted to graph classes closed under induced subgraphs and disjoint unions, Independent Set is polynomial-time solvable if both input interval graphs are cluster graphs, and is $\mathrm{NP}$-hard otherwise.

• 2. We chart the possibilities and limits of effective polynomial-time preprocessing (also known as kernelization).

• 3. We extend Halldórsson and Karlsson (2006)’s fixed-parameter algorithm for Independent Set on strip graphs parameterized by the structural parameter “maximum number of live jobs” to show that the problem (also known as Job Interval Selection) is fixed-parameter tractable with respect to the parameter $k$ and generalize their algorithm from strip graphs to 2-union graphs. Preliminary experiments with random data indicate that Job Interval Selection with up to 15 jobs and $5\cdot 10^5$ intervals can be solved optimally in less than 5 min.