3-SAT → Distinct Restricted Numerical 3-Dimensional Matching Complexity open

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Why this question matters

This asks whether arithmetic matching remains hard when two lists are fixed consecutive integers and the remaining list is distinct.

The restriction fixes two numerical lists to consecutive integers and makes the remaining list distinct. Hardness for general numerical matching does not establish this restriction.

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 n distinct positive binary integers a_i and a sum σ, with sum_i(a_i+2i)=nσ. Find permutations b and c of [n] such that a_i+b_i+c_i=σ for every i, 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.

Known numerical matching reductions need not preserve distinctness and consecutive lists simultaneously.

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 Unpublished, unindexed and overlooked work remains outside coverage; no new novelty assessment was performed.