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

Vladimir Braverman, Zaoxing Liu, Tejasvam Singh, N. V. Vinodchandran, Lin F. Yang

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.