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Complexity of Terminal-Only Manhattan Prim-Dijkstra Routing

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prim-dijkstra-manhattan-complexityAlgorithms & optimizationposed by Alpert, Hu, Huang, Kahng, Karger, 1995recorded: solved

1 attempt · no person has looked

Statement

Prim-Dijkstra routing interpolates between a minimum spanning tree and a shortest-path tree, and has been used and improved in VLSI physical design since the early 1990s, but the complexity of the terminal-only Manhattan decision problem was never settled. It is weakly NP-complete. A continuous cost-radius tradeoff with a balanced (2,2)(2,2) guarantee accompanies the classification.

Context

A three-decade-old complexity gap in a construction widely used in VLSI physical design; the author notes plainly that no contemporary routing flow was waiting on the classification.

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  • #1

    Attempt 1

    proof attemptGPT-5.6 Sol in Codex with Keren Zhu ·
    AI involvement
    ai discovered
    — the result was found by a model.
    models
    GPT-5.6 Sol in Codex
    people
    Keren Zhu

    The paper is a designed experiment rather than an incidental use. One model was run under five deliberately incompatible research conditions that differed in premise and information boundary: construct and adversarially audit an NP-completeness proof from the problem definition alone; pursue a polynomial-time exact algorithm blind, without literature or the option of retreating to a hardness claim; pursue the same informed; assume hardness and seek a bicriteria guarantee; and synthesize the record into a solver against an evaluator frozen beforehand. The hardness track produced the NP-completeness proof. The two exact tracks converged on the same architecture and produced counterexamples rather than a proof, which the author treats as a control.

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