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Large Clique is Hard on Average for Resolution

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Computer Science – Theory and Applications (CSR 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12730))

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Abstract

We prove an \(\exp ({\varOmega (k^{(1-\epsilon )})})\) resolution size lower bound for the k-Clique problem on random graphs, for (roughly speaking) \(k<n^{1/3}\). Towards an optimal resolution lower bound of the problem (i.e. of type \(n^{\varOmega (k)}\)), we also extend the \(n^{\varOmega (k)}\) bound in [2] on regular resolution to a stronger model called a-irregular resolution, for \(k<n^{1/3}\). This model is interesting in that all known CNF families separating regular resolution from general [1, 24] have short proofs in it.

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Notes

  1. 1.

    More precisely, it is a product-subset of \([n/k]^k\); the reason is clear after seeing the strong encoding (Sect. 2) where the vertex set is partitioned into k parts.

  2. 2.

    There is also the so-called binary encoding [15], which we will not discuss here.

  3. 3.

    For complete \((k-1)\)-partite graphs, a similar reduction is observed earlier by Alexander Razborov (personal communication).

  4. 4.

    For some specially structured G this is possible; see Remark 2.

  5. 5.

    Although a variant of it seems sufficient for tree-like resolution, cf. [5].

  6. 6.

    The actual construction has one more twist called mirroring, which we ignore here.

  7. 7.

    Other partitions might also seem natural but fail the second property, and we do not know the power of the model with them.

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Acknowledgment

I am indebted to Alexander Razborov for many helpful communications and feedback on the early draft. My thanks also go to Aaron Potechin, Jakob Nordström, and Ilario Bonacina, for various comments and references, and to the anonymous referees for their extensive feedback and suggestions that undoubtedly help improve readability.

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  1. Mathematics Department, University of Chicago, Chicago, USA

    Shuo Pang

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  1. Shuo Pang
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Correspondence to Shuo Pang .

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Editors and Affiliations

  1. University of Oxford, Oxford, UK

    Rahul Santhanam

  2. Innovations & HighTech, Moscow Institute of Physics and Technology, Moscow, Russia

    Daniil Musatov

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Pang, S. (2021). Large Clique is Hard on Average for Resolution. In: Santhanam, R., Musatov, D. (eds) Computer Science – Theory and Applications. CSR 2021. Lecture Notes in Computer Science(), vol 12730. Springer, Cham. https://doi.org/10.1007/978-3-030-79416-3_22

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  • DOI: https://doi.org/10.1007/978-3-030-79416-3_22

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