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Counting all independent sets and vertex covers is one-call #P-complete

Lax859101.AllSetsComplete · concepts/Lax859101/AllSetsComplete.lean · lax-859101

proven

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    Natural Language Statement

    Theorem

    Counting all the independent sets of a graph is one-call #P-complete, by a one-call reduction from #Independent Set, and so is counting all the vertex covers, which are the complements of the independent sets.

    Concept map
    22 concepts
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 2 statements. Each proof establishes one of them relative to its assumptions.

    1 sharpAllIndependentSets_sharpP_oneCallComplete proven

    2 sharpAllVertexCovers_sharpP_oneCallComplete proven

    Lean source view on GitHub

    1import Lax904597.Problems
    2import Lax904597.Sat
    3import Mathlib.ModelTheory.Graph
    4import Lax799700.SetFamily
    5import Lax366625.CountingProblems
    6import Lax366625.CountingClasses
    7import Lax366625.WitnessCounting
    8import Lax366625.CountingSat
    9import Lax859101.OneCallReductions
    10import Lax859101.SubtractiveReductions
    11import Lax859101.CountingDnf
    12import Lax859101.CountingNaeSat
    13import Lax859101.CountingRestrictedSat
    14import Lax859101.CountingAllSets
    15import Lax859101.CountingBipartite
    16
    17/-!
    18---
    19title: Counting all independent sets and vertex covers is one-call #P-complete
    20type: theorem
    21---
    22Counting all the independent sets of a graph is one-call #P-complete, by a
    23one-call reduction from #Independent Set, and so is counting all the vertex
    24covers, which are the complements of the independent sets.
    25-/
    26
    27namespace Lax859101.AllSetsComplete
    28
    29open FirstOrder FirstOrder.Language FirstOrder.Language.Structure
    30open Lax904597.Problems Lax904597.Sat Lax799700.SetFamily
    31open Lax366625.CountingProblems Lax366625.CountingClasses Lax366625.WitnessCounting
    32 Lax366625.CountingSat
    33open Lax859101.OneCallReductions Lax859101.SubtractiveReductions Lax859101.CountingDnf
    34 Lax859101.CountingNaeSat Lax859101.CountingRestrictedSat Lax859101.CountingAllSets
    35 Lax859101.CountingBipartite
    36
    37/-- Counting all independent sets is one-call #P-complete. -/
    38axiom sharpAllIndependentSets_sharpP_oneCallComplete :
    39 OneCallComplete SharpP SharpAllIndependentSets
    40
    41/-- Counting all vertex covers is one-call #P-complete. -/
    42axiom sharpAllVertexCovers_sharpP_oneCallComplete :
    43 OneCallComplete SharpP SharpAllVertexCovers
    44
    45end Lax859101.AllSetsComplete
    46
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