3-SAT → Constant-free graph orientation Complexity open

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Clone https://github.com/reduction-zoo/constant-free-orientation to investigate 3-SAT → Constant-free graph orientation. Read the fixed question and state. Use preparation notes, the test corpus, and verifier if present; otherwise complete Prepare before a reduction attempt. Follow the repository instructions and your AutoResearch pipeline.

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Why this question matters

This tests whether a small uniform orientation constraint can encode Boolean computation without supplied constants.

A reduction that supplies fixed port values would establish a different result. The question here permits only the two stated local indegree rules.

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 is a graph allowing parallel edges and loops. Degree-eight vertices require indegree in {1,4}; degree-two vertices require indegree in {0,2}. A solution orients every edge subject to these rules, without externally fixed ports.

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.

The source formulation and constant-forcing mechanism require care; a reduction with preassigned ports does not settle this question.

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 2026-09-16. This summarizes the archived literature search on the date above. Unpublished, unindexed and overlooked work remains outside coverage; no new novelty assessment was performed.