Lax Lean Archive

While this submission is a draft, it cannot be used by other submissions.

Håstad inapproximability for bounded-error randomized algorithms

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

proven

Loading review…

Community review

Flags

Each flag is tied to a public ORCID identity and explains why this concept may be incorrect.

No flags have been submitted.

    Community review

    Flag this concept

    State precisely what appears incorrect. This explanation will be public under your ORCID name.

    No source line selected.

    Natural Language Statement

    Theorem

    For every fixed ε>0\varepsilon>0, a polynomial-time randomized n1−εn^{1-\varepsilon} approximation of the clique number, successful with probability at least 2/32/3 on every nonempty graph, implies NP⊆BPP\mathrm{NP}\subseteq\mathrm{BPP}. Thus NP⊈BPP\mathrm{NP}\nsubseteq\mathrm{BPP} rules out such randomized algorithms. The estimator may err in either direction on unsuccessful runs. NP and BPP are the registered Lax classes.

    Concept map
    10 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.RandomizedApproximation
    2import Lax434930.NondeterministicPolynomialTime
    3import Lax666725.RandomizedPolynomialTime
    4
    5/-!
    6---
    7title: Håstad inapproximability for bounded-error randomized algorithms
    8type: theorem
    9---
    10For every fixed ε>0\varepsilon>0, a polynomial-time randomized
    11n1−εn^{1-\varepsilon} approximation of the clique number, successful with
    12probability at least 2/32/3 on every nonempty graph, implies
    13NP⊆BPP\mathrm{NP}\subseteq\mathrm{BPP}. Thus NP⊈BPP\mathrm{NP}\nsubseteq\mathrm{BPP}
    14rules out such randomized algorithms. The estimator may err in either
    15direction on unsuccessful runs. NP and BPP are the registered Lax classes.
    16-/
    17
    18namespace Lax253009.RandomizedCliqueHardness
    19
    20open RandomizedApproximation Lax434930.NondeterministicPolynomialTime
    21open Lax666725.RandomizedPolynomialTime
    22
    23axiom approximation_implies_np_subset_bpp (ε : ℝ) (hε : 0 < ε) :
    24 Approximable ε → NP ⊆ BPP
    25
    26axiom not_approximable (ε : ℝ) (hε : 0 < ε) (hnot : ¬ NP ⊆ BPP) :
    27 ¬ Approximable ε
    28
    29end Lax253009.RandomizedCliqueHardness
    30
    Show ProofShow Proof
    Builds on
    Used by

    none

    From Mathlib

    none

    Discussion

    Ask a question or add context. Endorsements and structured flags are kept in the review panel above.

    Loading discussion…