Deterministic Lower Bounds for $k$-Edge Connectivity in the Distributed Sketching Model

TL;DR

This work establishes the first deterministic lower bound for a graph connectivity decision problem in the distributed sketching model by proving that any deterministic protocol deciding $k$-edge connectivity requires $\Omega(k)$ bits of communication in the worst case, for super-constant $k=\omega(1)$ up to $\gamma\sqrt{n}$. The authors introduce a new lower-bound graph family $\mathcal{G}_{k,n}$ and a 3-party problem, $\mathsf{UniqueOverlap}$, which capture the core difficulty and resist reductions from standard problems due to input correlations. A key technical achievement is a 0-error simulation in a three-party communication setting that transfers the hardness of $\mathsf{UniqueOverlap}$ to the distributed sketching model, avoiding probabilistic errors typical in prior reductions. The result closes a long-standing question about deterministic graph sketching by showing near-linear lower bounds and opens avenues for further exploration of deterministic versus randomized bounds in this model. The techniques—new graph constructions, cross-intersecting set-family arguments, and a deterministic simulation—may influence future deterministic lower bounds in distributed computation and sketching.

Abstract

We study the $k$-edge connectivity problem on undirected graphs in the distributed sketching model, where we have $n$ nodes and a referee. Each node sends a single message to the referee based on its 1-hop neighborhood in the graph, and the referee must decide whether the graph is $k$-edge connected by taking into account the received messages. We present the first lower bound for deciding a graph connectivity problem in this model with a deterministic algorithm. Concretely, we show that the worst case message length is $Ω( k )$ bits for $k$-edge connectivity, for any super-constant $k = O(\sqrt{n})$. Previously, only a lower bound of $Ω( \log^3 n )$ bits was known for ($1$-edge) connectivity, due to Yu (SODA 2021). In fact, our result is the first super-polylogarithmic lower bound for a connectivity decision problem in the distributed graph sketching model. To obtain our result, we introduce a new lower bound graph construction, as well as a new 3-party communication complexity problem that we call UniqueOverlap. As this problem does not appear to be amenable to reductions to existing hard problems such as set disjointness or indexing due to correlations between the inputs of the three players, we leverage results from cross-intersecting set families to prove the hardness of UniqueOverlap for deterministic algorithms. Finally, we obtain the sought lower bound for deciding $k$-edge connectivity via a novel simulation argument that, in contrast to previous works, does not introduce any probability of error and thus works for deterministic algorithms.

Theorems & Definitions (24)

• Theorem 1
• Theorem 2
• Lemma 1
• Lemma 2
• proof
• Definition 1: Separated Pair
• Lemma 3
• Definition 2: Indistinguishable Separated Pair
• Lemma 4
• proof