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

TL;DR

This paper investigates the parameterized complexity of scheduling unit-time jobs with generalized Boolean formula-based precedence constraints, revealing fixed-parameter tractability in some cases and hardness in others.

Contribution

It introduces a unified framework for various precedence constraints and characterizes the complexity landscape based on different parameterizations and constraint structures.

Findings

Fixed-parameter tractability when parameterizing by the number of predecessors.

W[1]-hardness for disjunctive normal form constraints.

NP-hardness results for and/or-constrained problems and on two machines.

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).