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Faster zero-weight k-clique

Lax350013.ZeroWeightClique · concepts/Lax350013/ZeroWeightClique.lean · lax-350013

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    Natural Language Statement

    Theorem

    For every fixed k≥3k\geq 3, a zero-weight clique with one vertex in each of kk parts of size nn is decidable deterministically in O(nk−0.0017⌊k/3⌋)O(n^{k-0.0017\lfloor k/3\rfloor}) word-RAM steps. Edge weights are polynomially bounded integers (Corollary 39).

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

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    1/-
    2Copyright (c) 2026 Anthropic, PBC. All rights reserved.
    3Released under Apache 2.0 license as described in the file LICENSE.
    4SPDX-License-Identifier: Apache-2.0
    5-/
    6/-
    7Modified for the independent Lax packaging by Édouard Bonnet, 2026.
    8Derived from 3sum-apsp/EndStatement.lean / PaperStatements.lean at upstream commit e1a4e6508154ea59f030480661590a9fe3018011.
    9Changes: Lax module/namespace layout, separated concepts and proofs, archive
    10annotations, and compatibility with the archive Lean/mathlib environment.
    11See NOTICE and README.md in the submission root for provenance and scope.
    12-/
    13
    14import Lax350013.PolynomialTime
    15import Lax350013.ExactTriangle
    16
    17/-!
    18---
    19title: Faster zero-weight k-clique
    20type: theorem
    21---
    22For every fixed k≥3k\geq 3, a zero-weight clique with one vertex in each of kk parts of size nn is decidable deterministically in O(nk−0.0017⌊k/3⌋)O(n^{k-0.0017\lfloor k/3\rfloor}) word-RAM steps. Edge weights are polynomially bounded integers (Corollary 39).
    23-/
    24
    25namespace Lax350013.ZeroWeightClique
    26
    27open Lax350013.PolynomialTime
    28open Lax350013.ExactTriangle
    29
    30/-- `w i j u v`: weight between `u` in part `i` and `v` in part `j`. All `k²` blocks are input; only `i < j` counts. -/
    31def ZeroWeightKClique (k : Nat) : Problem where
    32 Instance n := Fin k → Fin k → Fin n → Fin n → Int
    33 input w := (List.ofFn fun i => (List.ofFn fun j => rowByRow (w i j)).flatten).flatten
    34 yes {n} w := ∃ v : Fin k → Fin n,
    35 (List.ofFn fun j => (List.ofFn fun i => if i < j then w i j (v i) (v j) else 0).sum).sum = 0
    36
    37/-- Corollary 39: «decide in O(n^(k−ε_T⌊k/3⌋)) time whether some k-clique … has total edge weight zero». -/
    38def Corollary_39_ZeroWeight : Prop :=
    39 ∀ k ≥ 3, (ZeroWeightKClique k).SolvedInTime (k - ε_T * (k / 3 : Nat))
    40
    41/-- Faster zero-weight k-clique: Corollary 39 ZeroWeight. -/
    42axiom algorithm : Corollary_39_ZeroWeight
    43
    44end Lax350013.ZeroWeightClique
    45
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