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Clique inapproximability under NP not contained in BPP

Lax253009.BPPConsequence · concepts/Lax253009/BPPConsequence.lean · lax-253009

proven

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    Natural Language Statement

    Theorem

    For every fixed ε>0\varepsilon>0, a polynomial-time n1−εn^{1-\varepsilon}-approximation of the clique number would imply NP⊆BPP\mathrm{NP}\subseteq\mathrm{BPP}. Consequently, the assumption NP⊈BPP\mathrm{NP}\nsubseteq\mathrm{BPP} rules out such an approximation.

    The proof uses Håstad's NP = ZPP implication and the ZPP ⊆ BPP inclusion from lax-666725. Both dependencies have proofs, so the archive dependency closure also proves this consequence.

    Concept map
    11 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 approximation_implies_np_subset_bpp proven

    2 not_approximable proven

    Lean source view on GitHub

    1import Lax253009.CliqueHardness
    2import Lax666725.RandomizedPolynomialTime
    3import Lax666725.ZPPSubsetBPP
    4
    5/-!
    6---
    7title: Clique inapproximability under NP not contained in BPP
    8type: theorem
    9---
    10For every fixed ε>0\varepsilon>0, a polynomial-time
    11n1−εn^{1-\varepsilon}-approximation of the clique number would imply
    12NP⊆BPP\mathrm{NP}\subseteq\mathrm{BPP}. Consequently, the assumption
    13NP⊈BPP\mathrm{NP}\nsubseteq\mathrm{BPP} rules out such an approximation.
    14
    15The proof uses Håstad's NP = ZPP implication and the ZPP ⊆ BPP inclusion
    16from lax-666725. Both dependencies have proofs, so the archive dependency
    17closure also proves this consequence.
    18-/
    19
    20namespace Lax253009.BPPConsequence
    21
    22open Approximation
    23open Lax434930.NondeterministicPolynomialTime
    24open Lax666725.RandomizedPolynomialTime
    25
    26axiom approximation_implies_np_subset_bpp (ε : ℝ) (hε : 0 < ε) :
    27 Approximable ε → NP ⊆ BPP
    28
    29axiom not_approximable (ε : ℝ) (hε : 0 < ε) (hnot : ¬ NP ⊆ BPP) :
    30 ¬ Approximable ε
    31
    32end Lax253009.BPPConsequence
    33
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