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):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 =Õ(log(n)), we prove that the space complexity of CLIQUE-GAP problem is˜ ( 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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