Deterministic near-optimal distributed listing of cliques

The importance of classifying connections in large graphs has been the motivation for a rich line of work on distributed subgraph finding that has led to exciting recent breakthroughs. A crucial aspect that remained open was whether deterministic algorithms can be as efficient as their randomized counterparts, where the latter are known to be tight up to polylogarithmic factors. We give deterministic distributed algorithms for listing cliques of size p in \(n^{1 - 2/p + o(1)}\) $n^{1-2/p+o(1)}$ rounds in the Congest model. For triangles, our \(n^{1/3+o(1)}\) $n^{1/3+o(1)}$ round complexity improves upon the previous state of the art of \(n^{2/3+o(1)}\) $n^{2/3+o(1)}$ rounds (Chang and Saranurak, in: 2020 IEEE 61st annual symposium on foundations of computer science (FOCS), pp 377–388. IEEE Computer Society, Los Alamito, 2020. https://doi.org/10.1109/FOCS46700.2020.00043). For cliques of size \(p \ge 4\) $p\ge 4$ , ours are the first non-trivial deterministic distributed algorithms. Given known lower bounds, for all values \(p \ge 3\) $p\ge 3$ our algorithms are tight up to an \(n^{o(1)}\) $n^{o(1)}$ subpolynomial factor, which comes from the deterministic routing procedure we use.