Concurrency Constrained Scheduling with Tree-Like Constraints

2025-04-30

AI Summary

This paper studies job scheduling under concurrency constraints: given a conflict graph (where edges encode incompatibility between jobs) and job attributes (processing times, deadlines, weights), the goal is to optimize classical scheduling objectives—such as weighted sum of completion times or maximum lateness—while respecting concurrency restrictions. Since the problem is NP-hard on general graphs, the work focuses on graphs of bounded treewidth and establishes, for the first time, a treewidth-parameterized complexity dichotomy: it devises fixed-parameter tractable (FPT) algorithms for multiple objectives and proves XALP-completeness for several variants. A key innovation is the introduction of the Grundy number to tightly bound makespan, moving beyond traditional graph coloring frameworks. The approach unifies and generalizes classical results—including chromatic number and sum coloring—for bounded-treewidth graphs, yielding a tight parameterized characterization of tractability for resource-competitive scheduling.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

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.

Problem

• Scheduling jobs with concurrency constraints on conflict graphs
• Optimizing schedules under tree-like constraints and NP-hard conditions
• Exploring FPT algorithms for bounded treewidth conflict graphs

Innovation

• Scheduling with tree-like concurrency constraints
• FPT algorithms for bounded treewidth graphs
• Grundy number bounds completion times