Set Splitting → Binary-network common contraction Complexity open

Model and version: gpt-6-astra
Date: Not recorded

Solution submitted Research history & proof on GitHub ↗

Why this question matters

This isolates the difficulty of comparing evolutionary networks under a basic structural simplification.

Hardness for unrestricted network contraction does not immediately apply when both input networks must be binary.

The reduction to construct

The source gives a finite ground set and an explicit family of subsets. A valid output assigns two colors so that every subset meets both colors, or reports NO-SOLUTION.

The target gives two binary rooted acyclic networks on the same uniquely labelled leaves and a bound k. Find a common contraction with at least k internal vertices, using acyclicity-preserving internal-edge contractions, or report NO-SOLUTION. A witness supplies both contraction sequences and their labelled isomorphism.

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.

Soundness must cover arbitrary contraction sequences, including merges outside intended gadgets.

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 Unpublished, unindexed and overlooked work remains outside coverage; no new novelty assessment was performed.