3-SAT → Dimension of a poset with a planar diagram Complexity open
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Why this question matters
This separates the complexity of order dimension from the complexity of obtaining a planar drawing.
Supplying a planar diagram does not fix a realizer. The proposed result concerns dimension despite that additional geometric information.
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 supplies a finite poset, an upward planar Hasse diagram with rational polyline coordinates, and a dimension bound d. Find at most d linear orders whose intersection is the poset, 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.
The reduction must construct a valid upward planar diagram while enforcing every order constraint.
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 Unpublished, unindexed and overlooked work remains outside coverage; no new novelty assessment was performed.