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New Bounds for the CLIQUE-GAP Problem Using Graph Decomposition Theory

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Abstract

Halldórsson et al (ICALP proceedings of the 39th international colloquium conference on automata, languages, and programming, vol part I, Springer, pp 449–460, 2012) investigated the space complexity of the following problem CLIQUE-GAP(r, s): given a graph stream G, distinguish whether \(\omega (G) \ge r\) or \(\omega (G) \le s\), where \(\omega (G)\) is the clique-number of G. In particular, they give matching upper and lower bounds for CLIQUE-GAP(r, s) for any r and \(s =c\log (n)\), for some constant c. The space complexity of the CLIQUE-GAP problem for smaller values of s is left as an open question. In this paper, we answer this open question. Specifically, for any r and for \(s={\tilde{O}}(\log (n))\), we prove that the space complexity of CLIQUE-GAP problem is \({\tilde{\Theta }}(\frac{ms^2}{r ^2})\). Our lower bound is based on a new connection between graph decomposition theory (Chung et al in Studies in pure mathematics, Birkhäuser, Basel, pp 95–101, 1983; Chung in SIAM J Algebr Discrete Methods 2(1):1–12, 1981) and the multi-party set disjointness problem in communication complexity.

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Notes

  1. In this and following theorems, the constants we choose are only for demonstrative convenience.

  2. Note that some papers define the decomposition on connected graph. We here use a more general statement.

  3. After the preliminary version on MFCS 2015 [17], McGregor et al. [24] give a two-pass algorithm of \(O(m^{3/2}/T)\) memory on the incident model.

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Authors and Affiliations

  1. Johns Hopkins University, Baltimore, MD, 21218, USA

    Vladimir Braverman, Zaoxing Liu, Tejasvam Singh & Lin F. Yang

  2. University of Nebraska–Lincoln, Lincoln, NE, 68588, USA

    N. V. Vinodchandran

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  1. Vladimir Braverman
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  2. Zaoxing Liu
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  3. Tejasvam Singh
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  4. N. V. Vinodchandran
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  5. Lin F. Yang
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Corresponding author

Correspondence to Zaoxing Liu.

Additional information

Vladimir Braverman: This material is based upon work supported in part by the National Science Foundation under Grant No. 1447639, by the Google Faculty Award and by DARPA Grant N660001-1-2-4014. Its contents are solely the responsibility of the authors and do not represent the official view of DARPA or the Department of Defense.

Zaoxing Liu: This work is supported in part by DARPA Grant N660001-1-2-4014.

N. V. Vinodchandran: Research supported in part by National Science Foundation Grant CCF-1422668.

Lin F. Yang: This work is supported in part by NSF Grant No. 1447639.

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Braverman, V., Liu, Z., Singh, T. et al. New Bounds for the CLIQUE-GAP Problem Using Graph Decomposition Theory. Algorithmica 80, 652–667 (2018). https://doi.org/10.1007/s00453-017-0277-5

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