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#Exact Cover is parsimoniously #P-complete

Lax280166.ExactCoverComplete · concepts/Lax280166/ExactCoverComplete.lean · lax-280166

proven

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    Natural Language Statement

    Theorem

    #Exact Cover is parsimoniously #P-complete: it is in #P, and every problem of #P reduces to it by a relativized ordered parsimonious reduction. Hardness comes from #1-in-SAT by a parsimonious reduction. The support of #Exact Cover, the instances with a positive count, is the decision problem ExactCover of the NP catalog.

    Concept map
    32 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 sharpExactCover_sharpP_parsimoniousComplete proven

    2 sharpExactCover_support_iff proven

    Lean source view on GitHub

    1import Lax904597.Problems
    2import Lax904597.Sat
    3import Lax799700.CliqueFamily
    4import Lax799700.DominatingSet
    5import Lax799700.Feedback
    6import Lax799700.Hamilton
    7import Lax799700.Knapsack
    8import Lax799700.OneInSat
    9import Lax799700.SetFamily
    10import Lax799700.Steiner
    11import Lax799700.ThreeSat
    12import Lax799700.ZeroOneIP
    13import Lax366625.CountingProblems
    14import Lax366625.CountingClasses
    15import Lax366625.WitnessCounting
    16import Lax366625.CountingSat
    17import Lax280166.CountingSatVariants
    18import Lax280166.CountingCliques
    19import Lax280166.CountingDominatingSets
    20import Lax280166.CountingFeedbackSets
    21import Lax280166.CountingHamiltonCircuits
    22import Lax280166.CountingSetFamilies
    23import Lax280166.CountingKnapsacks
    24import Lax280166.CountingSteinerTrees
    25
    26/-!
    27---
    28title: #Exact Cover is parsimoniously #P-complete
    29type: theorem
    30---
    31#Exact Cover is parsimoniously #P-complete: it is in #P, and every problem of #P
    32reduces to it by a relativized ordered parsimonious reduction. Hardness
    33comes from #1-in-SAT by a parsimonious reduction.
    34The support of #Exact Cover, the instances with a positive count, is the decision
    35problem ExactCover of the NP catalog.
    36-/
    37
    38namespace Lax280166.ExactCoverComplete
    39
    40open FirstOrder FirstOrder.Language
    41open Lax904597.Problems Lax904597.Sat
    42open Lax366625.CountingProblems Lax366625.CountingClasses Lax366625.WitnessCounting
    43 Lax366625.CountingSat
    44open Lax799700.ThreeSat Lax799700.SetFamily
    45open Lax280166.CountingSatVariants Lax280166.CountingCliques Lax280166.CountingDominatingSets
    46 Lax280166.CountingFeedbackSets Lax280166.CountingHamiltonCircuits Lax280166.CountingSetFamilies
    47 Lax280166.CountingKnapsacks Lax280166.CountingSteinerTrees
    48
    49/-- #Exact Cover is parsimoniously #P-complete. -/
    50axiom sharpExactCover_sharpP_parsimoniousComplete :
    51 SharpP.ParsimoniousComplete SharpExactCover
    52
    53/-- The support of #Exact Cover is the decision problem ExactCover. -/
    54axiom sharpExactCover_support_iff :
    55 ∀ (A : Type) [Lax799700.SetFamily.setSystem.Structure A] [Finite A],
    56 SharpExactCover.support A ↔ Lax799700.SetFamily.ExactCover A
    57
    58end Lax280166.ExactCoverComplete
    59
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