Superlinear Lower Bounds for Distributed Subgraph Detection

TL;DR

This paper proves that detecting certain subgraphs in distributed networks with bandwidth constraints requires superlinear time, establishing new lower bounds that improve understanding of the problem's inherent complexity.

Contribution

It introduces superlinear lower bounds for distributed subgraph detection, showing the problem's difficulty exceeds previous linear or sublinear bounds for specific subgraphs.

Findings

Superlinear lower bounds for subgraph detection in distributed networks.

Lower bounds depend on subgraph size and network bandwidth.

Detection of certain subgraphs requires (n^{2-1/k}/(Bk)) rounds, even for small diameters.

Abstract

In the distributed subgraph-freeness problem, we are given a graph $H$, and asked to determine whether the network graph contains $H$ as a subgraph or not. Subgraph-freeness is an extremely local problem: if the network had no bandwidth constraints, we could detect any subgraph $H$ in $|H|$ rounds, by having each node of the network learn its entire $|H|$-neighborhood. However, when bandwidth is limited, the problem becomes harder. Upper and lower bounds in the presence of congestion have been established for several classes of subgraphs, including cycles, trees, and more complicated subgraphs. All bounds shown so far have been linear or sublinear. We show that the subgraph-freeness problem is not, in general, solvable in linear time: for any $k \geq 2$, there exists a subgraph $H_k$ such that $H_k$-freeness requires $\Omega( n^{2-1/k} / (Bk) )$ rounds to solve. Here $B$ is the bandwidth of each communication link. The lower bound holds even for diameter-3 subgraphs and diameter-3 network graphs. In particular, taking $k = \Theta(\log n)$, we obtain a lower bound of $\Omega(n^2 / (B \log n))$.