Counting $t$-cliques: Worst-case to average-case reductions and Direct interactive proof systems.

We study two aspects of the complexity of counting the number of t-cliques in a graph: 1) Worst-case to average-case reductions: Our main result reduces counting t-cliques in any n-vertex graph to counting t-cliques in typical n-vertex graphs that are drawn from a simple distribution of min-entropy Ω(n2). For any constant t, the reduction runs in O(n2)-time, and yields a correct answer (w.h.p.) even when the “average-case solver” only succeeds with probability 1/poly(log n). 2) Direct interactive proof systems: We present a direct and simple interactive proof system for counting t-cliques in n-vertex graphs. The proof system uses t - 2 rounds, the verifier runs in O(t2n2)-time, and the prover can be implemented in O(tO(1) · n2)-time when given oracle access to counting (t - 1)-cliques in O(tO(1) · n)-vertex graphs. The results are both obtained by considering weighted versions of the t-clique problem, where weights are assigned to vertices and/or to edges, and the weight of cliques is defined as the product of the corresponding weights. These weighted problems are shown to be easily reducible to the unweighted problem.