How Proofs are Prepared at Camelot

We study a design framework for robust, independently verifiable, and workload-balanced distributed algorithms working on a common input. The framework builds on recent noninteractive Merlin--Arthur proofs of batch evaluation of Williams~[31st IEEE Colloquium on Computational Complexity (CCC'16, May 29-June 1, 2016, Tokyo), to appear] with the basic observation that Merlin's magic is not needed for batch evaluation: mere Knights can prepare the independently verifiable proof, in parallel, and with intrinsic error-correction. As our main technical result, we show that the k-cliques in an n-vertex graph can be counted and verified in per-node O(n(ω+ε)k/6) time and space on O(n(ω+ε)k/6) compute nodes, for any constant ε>0 and positive integer k divisible by 6, where 2 ≤ ω < 2.3728639 is the exponent of square matrix multiplication over the integers. This matches in total running time the best known sequential algorithm, due to Nešetřil and Poljak [Comment. Math. Univ. Carolin. 26 (1985) 415--419], and considerably improves its space usage and parallelizability. Further results (only partly presented in this extended abstract) include novel algorithms for counting triangles in sparse graphs, computing the chromatic polynomial of a graph, and computing the Tutte polynomial of a graph