3-SAT → Three-machine unit-job precedence scheduling Complexity open

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Why this question matters

This is a longstanding small-machine scheduling boundary with broad relevance to precedence-constrained computation.

The three-machine unit-job precedence case is a classical scheduling question; hardness when the number of machines varies does not resolve this fixed-machine case.

The reduction to construct

A source instance is an explicitly encoded Boolean formula with at most three literals per clause. Its outputs are satisfying Boolean assignments, or NO-SOLUTION when the formula is unsatisfiable.

The target gives a precedence DAG of unit-time jobs and an integer deadline T. Find an integer-slot schedule on three identical machines respecting precedences and finishing by T, or report NO-SOLUTION.

Construct deterministic polynomial-time maps F and G: F sends every legal source instance to a legal target instance, and G(x,y) is a valid source output for every valid output y of F(x). Preserve the stated threshold, domain and promises. The requested complexity conclusion is NP-hardness for the stated target problem.

With only three machines and unit jobs, gadgets have little numerical freedom and must enforce choices through precedences alone.

What a complete answer must include

Give explicit construction and recovery algorithms, a general proof for every legal input and every valid target output, and polynomial runtime and encoding-size bounds. Specify finite output encodings and handle NO-SOLUTION outputs when applicable. Tests compare recovered source outputs with independent source solutions.

References

Literature checked 2026-09-15. This summarizes the archived literature search on the date above. Unpublished, unindexed and overlooked work remains outside coverage; no new novelty assessment was performed.