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 G n, 1 2 (and hence is likely to have a clique of size roughly 2 log n), 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 n δ probes in ℓ rounds is likely (over the choice of the random graph) to output a clique of size larger than α log n.
Feige et al. REF show an impossibility result if the adaptivity of the algorithm is limited: any algorithm that makes q = O(n δ ) edge queries (δ < 2) in ℓ rounds finds cliques of size at most (2 − ε) log n where ε = ε (δ, ℓ) > 0.
Finding cliques using few probes
ArXiv
ArXiv
Mathematics Computer Science