Abstract
Halldórsson, Sun, Szegedy, and Wang (ICALP 2012) [16] 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 \(s=\tilde{O}(\log (n))\) and for any \(r>s\), we prove that the space complexity of CLIQUE-GAP problem is \(\tilde{\varTheta }(\frac{ms^2}{r^2})\). Our lower bound is based on a new connection between graph decomposition theory (Chung, Erdös, and Spencer [11], and Chung [10]) and the multi-party set disjointness problem in communication complexity.
V. 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.
Z. 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.
L.F. Yang—This work is supported in part by NSF grant No. 1447639.
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Notes
- 1.
In this and following theorems, the constants we choose are only for demonstrative convenience.
- 2.
In this and following theorems, the constants we choose are only for demonstrative convenience.
- 3.
Note that some papers define the decomposition on connected graph. We here use a more general statement.
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Braverman, V., Liu, Z., Singh, T., Vinodchandran, N.V., Yang, L.F. (2015). New Bounds for the CLIQUE-GAP Problem Using Graph Decomposition Theory. In: Italiano, G., Pighizzini, G., Sannella, D. (eds) Mathematical Foundations of Computer Science 2015. MFCS 2015. Lecture Notes in Computer Science(), vol 9235. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48054-0_13
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