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

TL;DR

This work studies scheduling unit-time jobs on identical parallel machines under generalized precedence constraints encoded as $\psi_j$ per job, aiming to minimize $C_{max}$ or the sum of completion times (and weights). It introduces two structural parameters, $k_p$ (predecessors) and $k_s$ (successors), and establishes an array of tractability results: $P \vert gen-prec, p_j=1 \vert \gamma$ is FPT in $k_p$ for $\gamma \in \{C_{max}, \sum_j C_j\}$, with the weighted case tractable when $k_p$ is bounded; for $k_s$ the outcome depends on constraint form, with $\text{and/+or}$-constraints giving FPT results, $\text{or/and}$ (DNF) yielding $W[1]$-hardness, and $\text{and/or}$ (CNF) being para-$\mathcal{NP}$-hard; two-machine cases exhibit NP-hardness (para-$\mathcal{NP}$-hardness in $m$) for mixed constraints. These results clarify when limiting dependencies yields tractability and identify hardness barriers for combined $\text{and}$/$\text{or}$ models, guiding both theory and practical scheduling. The work further provides algorithmic frameworks (e.g., predecessor- and successor-focused configurations) and suggests future directions on hybrid parameterizations and broader constraint families.

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).

Figures (4)

• Figure 1: We process six jobs $a,b,c,d,e,f$ with and+or-precedence constraints $\psi_e= x_a\land x_b\land x_c$, $\psi_f=x_a \lor x_d$ on three machines. The left schedule is optimal for the $and$-variation with $\psi'_f=x_d$. The schedule on the right is optimal for the $and$-variation with $\psi'_f=x_a$ and for the original instance.
• Figure 2: Reduction from Independent Set with $k=2$ to $P\vert or/and$-$prec, p_j=1 \vert \gamma$ with $k_s=2k+1$. Blue jobs and preceding orange jobs represent the size and selection of an independent set. In schedules with $C_{max}=2k+1$, green (red) jobs are used to block space after (before) blue jobs.
• Figure 3: Reduction of Vertex Cover with $n'=3$ and $k=1$ to $P|and/or$-$prec, p_j=1|\gamma$ with $k_s=1$
• Figure 4: Reduction from Vertex Cover to $P2|$and+or-prec$, p_j=1|\gamma$ with $J_s^{and}=\{b_6\}$.

Theorems & Definitions (33)

• Lemma 1
• proof
• Theorem 2
• proof
• Lemma 3
• proof
• Proposition 3
• proof
• Definition 4
• Lemma 4