Concurrency Constrained Scheduling with Tree-Like Constraints

This paper investigates concurrency-constrained scheduling problems, where the objective is to construct a schedule for a set of jobs subject to concurrency restrictions. Formally, we are given a conflict graph G defined over a set of n jobs, where an edge between two jobs in G indicates that these jobs cannot be executed concurrently. Each job may have distinct attributes, such as processing time, due date, weight, and release time. The goal is to determine a schedule that optimizes a specified scheduling criterion while adhering to all concurrency constraints. This framework offers a versatile model for analyzing resource allocation problems where processes compete for shared resources, such as access to shared memory. From a theoretical perspective, it encompasses several classical graph coloring problems, including Chromatic Number, Sum Coloring, and Interval Chromatic Number. Given that even the simplest concurrency-constrained scheduling problems are NP-hard for general conflict graphs, this study focuses on conflict graphs with bounded treewidth. Our results establish a dichotomy: Some problems in this setting can be solved in FPT time, while others are shown to be XALP-complete for treewidth as parameter. Along the way, we generalize several previously known results on coloring problems for bounded treewidth graphs. Several of the FPT algorithms are based on the insight that completion times are bounded by the Grundy number of the conflict graph—the fact that this number is bounded by the product of treewidth and the logarithm of the number of vertices then leads to the FPT time bound.