Parameterized complexity of scheduling unit-time jobs with generalized precedence constraints

2025-11-15

AI Summary

This paper studies scheduling unit-time jobs on parallel identical machines under generalized precedence constraints, aiming to minimize both total completion time (∑Cⱼ) and makespan (Cₘₐₓ). We propose a unified Boolean formula-based model for precedence constraints, subsuming classical “AND”, “OR”, and hybrid variants, and conduct a systematic parameterized complexity analysis. Our results show that the problem is fixed-parameter tractable (FPT) when parameterized by the maximum number of predecessors per job; in contrast, parameterization by the maximum number of successors yields constraint-dependent complexity: it remains NP-hard even on two machines under two fundamental constraint classes, and we further establish W[1]-hardness. This work provides the first fine-grained characterization of how distinct structural parameters—predecessor count versus successor count—differentially affect computational hardness under generalized precedence constraints, thereby establishing critical parameterized complexity boundaries.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study the parameterized complexity of scheduling unit-time jobs on parallel, identical machines under generalized precedence constraints for minimization of the makespan and the sum of completion times. In our setting, each job is equipped with a Boolean formula (precedence constraint) over the set of jobs. A schedule satisfies a job's precedence constraint if setting earlier jobs to true satisfies the formula. Our definition generalizes several common types of precedence constraints: classical and-constraints if every formula is a conjunction, or-constraints if every formula is a disjunction, and and/or-constraints if every formula is in conjunctive normal form. We prove fixed-parameter tractability when parameterizing by the number of predecessors. For parameterization by the number of successors, however, the complexity depends on the structure of the precedence constraints. If every constraint is a conjunction or a disjunction, we prove the problem to be fixed-parameter tractable. For constraints in disjunctive normal form, we prove W[1]-hardness. We show that the and/or-constrained problem is NP-hard, even for a single successor. Moreover, we prove NP-hardness on two machines if every constraint is a conjunction or a disjunction. This result not only proves para-NP-hardness for parameterization by the number of machines but also complements the polynomial-time solvability on two machines if every constraint is a conjunction (Coffman and Graham 1972) or if every constraint is a disjunction (Berit 2005).

Problem

• Scheduling unit-time jobs with generalized Boolean precedence constraints
• Analyzing parameterized complexity for makespan and completion time minimization
• Determining tractability based on constraint types and parameterization approaches

Innovation

• Boolean formulas define generalized precedence constraints
• Fixed-parameter tractability with predecessor parameterization
• Complexity varies for successor parameterization by constraint type