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

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 → 0 and N/M → ∞ we provide analytical results, while for intermediate values of N/M we perform experiments. We prove that as N/M → 0, backbone size approaches 100%, while as N/M → ∞ the backbone vanishes. In the process we show that as N/M → 0 (resp. N/M → ∞), 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.

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