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Lax47.Theorem12

Håstad hardness implies tight inapproximability in triangle-free graphs

concepts/Lax47/Theorem12.lean · lax-47

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    Theorem

    Assume Håstad's general-graph inapproximability premise. Then, for every constant ε>0\varepsilon>0, a polynomial-time N1/2−εN^{1/2-\varepsilon}-approximation algorithm for Max Independent Set on NN-vertex triangle-free graphs implies NP⊆BPPNP\subseteq BPP.

    The entire implication, including the randomized reduction from the Håstad promise gap, is the statement formalized below and proved by this submission. Its triangle-free conclusion is Theorem 1.2 in the submitted paper.

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    1import Lax47.Hastad
    2
    3/-!
    4---
    5title: Håstad hardness implies tight inapproximability in triangle-free graphs
    6type: theorem
    7---
    8Assume Håstad's general-graph inapproximability premise. Then, for every
    9constant ε>0\varepsilon>0, a polynomial-time
    10N1/2−εN^{1/2-\varepsilon}-approximation algorithm for Max Independent Set on
    11NN-vertex triangle-free graphs implies NP⊆BPPNP\subseteq BPP.
    12
    13The entire implication, including the randomized reduction from the Håstad
    14promise gap, is the statement formalized below and proved by this submission.
    15Its triangle-free conclusion is Theorem 1.2 in the submitted paper.
    16-/
    17
    18set_option autoImplicit false
    19
    20namespace Lax47.Theorem12
    21
    22open Lax47.Complexity
    23
    24/-- Håstad hardness implies tight conditional inapproximability on triangle-free graphs. -/
    25axiom theorem_1_2 :
    26 Lax47.Hastad.Inapproximability →
    27 ∀ (ε : ℝ), 0 < ε → TriangleFreeMISApproximation ε → NPSubsetBPP
    28
    29end Lax47.Theorem12
    30
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