[Paper Review] Scheduling Meets Fixed-Parameter Tractability

Matthias Mnich, Andreas Wiese|arXiv (Cornell University)|Nov 16, 2013

Scheduling and Optimization Algorithms25 references3 citations

TL;DR

This paper presents the first fixed-parameter tractable (FPT) algorithms for fundamental scheduling problems, including makespan minimization, scheduling with job-dependent cost functions (e.g., weighted flow time), and scheduling with rejection. By identifying the number of distinct processing times or rejection costs as a key parameter, the authors reduce these problems to integer programming in fixed dimension, enabling FPT algorithms despite numeric input values that typically hinder FPT approaches. The key contribution is establishing FPT algorithms for classical NP-hard scheduling problems under novel parameterizations, with matching W[1]-hardness results showing the tightness of these parameters.

ABSTRACT

Fixed-parameter tractability analysis and scheduling are two core domains of combinatorial optimization which led to deep understanding of many important algorithmic questions. However, even though fixed-parameter algorithms are appealing for many reasons, no such algorithms are known for many fundamental scheduling problems. In this paper we present the first fixed-parameter algorithms for classical scheduling problems such as makespan minimization, scheduling with job-dependent cost functions-one important example being weighted flow time-and scheduling with rejection. To this end, we identify crucial parameters that determine the problems' complexity. In particular, we manage to cope with the problem complexity stemming from numeric input values, such as job processing times, which is usually a core bottleneck in the design of fixed-parameter algorithms. We complement our algorithms with W[1]-hardness results showing that for smaller sets of parameters the respective problems do not allow FPT-algorithms. In particular, our positive and negative results for scheduling with rejection explore a research direction proposed by Dániel Marx. We hope that our contribution yields a new and fresh perspective on scheduling and fixed-parameter algorithms and will lead to further fruitful interdisciplinary research connecting these two areas.

Motivation & Objective

• To address the lack of fixed-parameter tractable (FPT) algorithms for classical scheduling problems despite their importance and practical relevance.
• To overcome the challenge posed by numeric input values (e.g., processing times, weights, rejection costs), which typically prevent FPT algorithms due to strong NP-hardness.
• To identify minimal, meaningful parameters—such as the number of distinct processing times or rejection costs—that render fundamental scheduling problems FPT.
• To establish both positive FPT results and W[1]-hardness lower bounds to show the tightness of the chosen parameters.
• To open a new interdisciplinary research direction connecting scheduling theory and fixed-parameter tractability.

Proposed method

• For makespan minimization on identical machines, the authors show that jobs of the same length can be nearly equally distributed across machines, enabling a preprocessing step that reduces the problem to a fixed-parameter form.
• The remaining scheduling decision is modeled as an integer program in fixed dimension, where variables represent the number of jobs of each size assigned to each machine.
• For unrelated machines, the parameterization includes the number of machines and the number of distinct processing times, with constraints derived from Hall’s theorem to ensure feasible job assignment.
• A compact set of Hall-type constraints is used to replace the exponentially many subset checks, enabling efficient FPT algorithms.
• For scheduling with rejection, the authors use a parameter based on the number of distinct rejection costs and reduce the problem to a variant of the knapsack problem with structured profit functions.
• The approach is extended to the General Scheduling Problem (GSP), proving NP-hardness even when processing times are restricted to 1 or 3, using a reduction from 3D-matching with binary encoding of job indices.

Experimental results

Research questions

• RQ1 Can classical scheduling problems such as makespan minimization and weighted flow time be solved in fixed-parameter tractable time when parameterized by the number of distinct processing times or job weights?
• RQ2 Is scheduling with rejection FPT when parameterized by the number of distinct rejection costs, despite the presence of numeric input values?
• RQ3 What is the minimal set of parameters that allows FPT algorithms for scheduling problems, and can we prove that smaller parameter sets lead to W[1]-hardness?
• RQ4 Can the number of distinct processing times serve as a sufficient parameter to achieve FPT algorithms for unrelated machine scheduling?
• RQ5 Is the General Scheduling Problem (GSP) NP-hard when processing times are restricted to only two values (1 and 3), even in the non-preemptive setting?

Key findings

• The paper presents the first FPT algorithm for makespan minimization on identical machines, parameterized by the number of distinct processing times, with a running time of f(k)·n^O(1).
• For unrelated machines, the problem R||C_max is FPT when parameterized by both the number of machines and the number of distinct processing times, via a fixed-dimension integer programming formulation.
• Scheduling with rejection is FPT when parameterized by the number of distinct rejection costs, with a reduction to a structured knapsack-like problem.
• The authors prove W[1]-hardness for all studied problems when parameterized by smaller parameter sets, showing the tightness of their chosen parameters.
• The General Scheduling Problem (GSP) is NP-hard even when all jobs have processing time 1 or 3, establishing a strong hardness result for a restricted but natural case.
• The paper establishes that solutions in standard form (non-preemptive, all jobs scheduled) can be used to reduce the problem to a parameterized setting, enabling FPT algorithms for both preemptive and non-preemptive variants.