Title:A Note on Improved Results for One Round Distributed Clique Listing

Abstract:In this note, we investigate listing cliques of arbitrary sizes in bandwidth-limited, dynamic networks. The problem of detecting and listing triangles and cliques was originally studied in great detail by Bonne and Censor-Hillel (ICALP 2019). We extend this study to dynamic graphs where more than one update may occur as well as resolve an open question posed by Bonne and Censor-Hillel (2019). Our algorithms and results are based on some simple observations about listing triangles under various settings and we show that we can list larger cliques using such facts. Specifically, we show that our techniques can be used to solve an open problem posed in the original paper: we show that detecting and listing cliques (of any size) can be done using $O(1)$-bandwidth after one round of communication under node insertions and node/edge deletions. We conclude with an extension of our techniques to obtain a small bandwidth $1$-round algorithm for listing cliques when more than one node insertion/deletion and/or edge deletion update occurs at any time.