Finding cliques using few probes

Abstract

© 2019 Wiley Periodicals, Inc. Consider algorithms with unbounded computation time that probe the entries of the adjacency matrix of an n vertex graph, and need to output a clique. We show that if the input graph is drawn at random from (Formula presented.) (and hence is likely to have a clique of size roughly (Formula presented.)), then for every δ<2 and constant ℓ, there is an α<2 (that may depend on δ and ℓ) such that no algorithm that makes nδ probes in ℓ rounds is likely (over the choice of the random graph) to output a clique of size larger than (Formula presented.).