3-SAT → Threshold-two Fitting Hexasort Complexity open
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Why this question matters
This asks whether the lowest nontrivial clearing threshold already supports computational hardness.
The threshold-two fitting question asks whether all stacks can be placed, rather than whether the final board can be cleared.
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 graph and a sequence of colored stacks. Each move places the next stack on an empty vertex; if it has a same-color neighbor, the new stack and its same-color neighbors clear. The goal is to place every stack, not to clear the final board.
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.
Every legal placement sequence must encode the source, despite clearing moves that can open unintended space.
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 Unpublished, unindexed and overlooked work remains outside coverage; no new novelty assessment was performed.