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Truly subcubic min-plus matrix multiplication

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

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    Natural Language Statement

    Theorem

    The min-plus product of two n×nn\times n matrices with polynomially bounded integer entries can be computed deterministically in O(n2.99942)O(n^{2.99942}) word-RAM steps (Theorem 22). The output contains the minimum of Aik+BkjA_{ik}+B_{kj} for each pair (i,j)(i,j).

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
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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
    15
    16/-!
    17---
    18title: Truly subcubic min-plus matrix multiplication
    19type: theorem
    20---
    21The min-plus product of two n×nn\times n matrices with polynomially bounded integer entries can be computed deterministically in O(n2.99942)O(n^{2.99942}) word-RAM steps (Theorem 22). The output contains the minimum of Aik+BkjA_{ik}+B_{kj} for each pair (i,j)(i,j).
    22-/
    23
    24namespace Lax350013.MinPlusProduct
    25
    26open Lax350013.PolynomialTime
    27
    28/-- Output, row by row: entry `(i, j)` is the least of the sums `A i k + B k j`. -/
    29def MinPlusProduct : Problem where
    30 Instance n := (Fin n → Fin n → Int) × (Fin n → Fin n → Int)
    31 input := fun (A, B) => rowByRow A ++ rowByRow B
    32 output := fun {n} (A, B) out => ∀ i j : Fin n,
    33 (∃ k, out (i.val * n + j.val) = A i k + B k j) ∧ ∀ k, out (i.val * n + j.val) ≤ A i k + B k j
    34
    35/-- Theorem 22: «the (min, +)-product of two n × n integer matrices», in time «O(n^2.99942)». -/
    36def Theorem_22_MinPlus : Prop :=
    37 MinPlusProduct.SolvedInTime 2.99942
    38
    39/-- Truly subcubic min-plus matrix multiplication: Theorem 22 MinPlus. -/
    40axiom algorithm : Theorem_22_MinPlus
    41
    42end Lax350013.MinPlusProduct
    43
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