How the Landscape of Random Job Shop Scheduling Instances Depends on the Ratio of Jobs to Machines

TL;DR

This paper analyzes how the structure of random job shop scheduling instances changes with the ratio of jobs to machines, revealing an 'easy-hard-easy' pattern in problem difficulty and providing both theoretical and experimental insights.

Contribution

It characterizes the search landscape of JSP instances across different job-machine ratios, including analytical results for extreme cases and experimental analysis for intermediate ratios.

Findings

Backbone size approaches 100% as N/M approaches 0.

Backbone vanishes as N/M approaches infinity.

Simple priority rules are nearly optimal in extreme cases.

Abstract

We characterize the search landscape of random instances of the job shop scheduling problem (JSP). Specifically, we investigate how the expected values of (1) backbone size, (2) distance between near-optimal schedules, and (3) makespan of random schedules vary as a function of the job to machine ratio (N/M). For the limiting cases N/M approaches 0 and N/M approaches infinity we provide analytical results, while for intermediate values of N/M we perform experiments. We prove that as N/M approaches 0, backbone size approaches 100%, while as N/M approaches infinity the backbone vanishes. In the process we show that as N/M approaches 0 (resp. N/M approaches infinity), simple priority rules almost surely generate an optimal schedule, providing theoretical evidence of an "easy-hard-easy" pattern of typical-case instance difficulty in job shop scheduling. We also draw connections between our theoretical results and the "big valley" picture of JSP landscapes.