The Communication Complexity of Set-Disjointness with Small Sets and 0-1 Intersection

In this paper, we analyze the following communication complexity problem. It is a variant of the set-disjointness problem, denoted$\PDISJ_{\log N}$, where each of Alice and Bob gets as an input a subset of $[N]$ of size at most $\log N$, with the promise that the intersection of the two subsets is of size at most $1$. We provide an almost tight lower bound of $\tOmega(\log^2 N)$ on the deterministic communication complexity of the problem. The main motivation for studying this problem comes from the so-called ''clique vs. independent-set'' problem, introduced by Yannakakis (1988). Proving an $\Omega(\log^2 N)$ lower bound on the communication complexity of the clique vs. independent-set problem for all graphs is a long standing open problem with various implications. Proving such a lower bound for {\em random}graphs is also open. In such a graph, both the cliques and the independent sets are of size $O(\log N)$ (and obviously their intersection is of size at most 1). Hence, our $\tOmega(\log^2 N)$lower bound for $\PDISJ_{\log N}$ can be viewed as a first step in this direction. Interestingly, we note that standard lower bound techniques cannot yield the desired lower bound. Hence, we develop a novel adversary argument that may find other applications.