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

Abstract

Halldorsson, 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, Erdos, and Spencer [11], and Chung [10]) and the multi-party set disjointness problem in communication complexity.