Lower Bounds for Induced Cycle Detection in Distributed Computing

TL;DR

This paper establishes tight and near-tight lower bounds for detecting induced cycles in distributed networks, revealing the increased complexity of induced subgraph detection compared to non-induced variants.

Contribution

It provides the first tight lower bounds for induced cycle detection in the CONGEST model, demonstrating the problem's higher complexity and extending understanding to larger cycles and specific subgraphs.

Findings

Induced 4-cycle detection requires rac{ ilde{ ext{O}}(n)}{ ext{rounds}} in CONGEST.

For cycles of length 5 to 7, the rac{ ilde{ ext{O}}(n)}{ ext{lower bound}} cannot be improved via communication complexity.

Induced cycle detection for k rac{ ext{at least}}{ ext{8}} requires rac{ ilde{ ext{O}}(n^{2- ext{Theta}(1/k)}) rounds, nearly matching upper bounds.

Abstract

The distributed subgraph detection asks, for a fixed graph $H$, whether the $n$-node input graph contains $H$ as a subgraph or not. In the standard CONGEST model of distributed computing, the complexity of clique/cycle detection and listing has received a lot of attention recently. In this paper we consider the induced variant of subgraph detection, where the goal is to decide whether the $n$-node input graph contains $H$ as an \emph{induced} subgraph or not. We first show a $\tilde{\Omega}(n)$ lower bound for detecting the existence of an induced $k$-cycle for any $k\geq 4$ in the CONGEST model. This lower bound is tight for $k=4$, and shows that the induced variant of $k$-cycle detection is much harder than the non-induced version. This lower bound is proved via a reduction from two-party communication complexity. We complement this result by showing that for $5\leq k\leq 7$, this $\tilde{\Omega}(n)$ lower bound cannot be improved via the two-party communication framework. We then show how to prove stronger lower bounds for larger values of $k$. More precisely, we show that detecting an induced $k$-cycle for any $k\geq 8$ requires $\tilde{\Omega}(n^{2-\Theta{(1/k)}})$ rounds in the CONGEST model, nearly matching the known upper bound $\tilde{O}(n^{2-\Theta{(1/k)}})$ of the general $k$-node subgraph detection (which also applies to the induced version) by Eden, Fiat, Fischer, Kuhn, and Oshman~[DISC 2019]. Finally, we investigate the case where $H$ is the diamond (the diamond is obtained by adding an edge to a 4-cycle, or equivalently removing an edge from a 4-clique), and show non-trivial upper and lower bounds on the complexity of the induced version of diamond detecting and listing.