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

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.

💡 Research Summary

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This paper investigates the deterministic communication complexity of deciding k‑edge connectivity in the distributed graph sketching model. In this model, each of the n nodes observes its 1‑hop neighborhood and sends a single message (a “sketch”) to a central referee, who must determine whether the input graph is k‑edge‑connected. While prior work established randomized lower bounds of Ω(log³ n) bits for connectivity and related problems, no super‑polylogarithmic lower bound was known for deterministic algorithms.

The authors prove that any deterministic sketching algorithm must use sketches of length Ω(k) bits in the worst case, for any super‑constant k ≤ γ√n (γ a constant). This is the first deterministic lower bound that grows linearly with k and exceeds polylogarithmic size.

To achieve this, the paper introduces two novel technical components:

1. A new hard graph construction – The authors design a tripartite graph (Vₗ, Vₘ, Vᵣ) similar to earlier constructions but with a distinguished vertex v_σ in Vₘ that has a slightly higher degree. The k‑edge‑connectivity of the entire graph hinges entirely on the choice of v_σ’s neighbors. All other vertices are constructed so that they are indistinguishable from v_σ’s perspective, making it impossible for a deterministic algorithm to infer the critical connectivity information without a sufficiently large sketch from v_σ.

2. The UniqueOverlap communication problem – This is a three‑party simultaneous message passing problem. Alice and Bob each hold an m‑length vector over {0,1,⊥} with exactly one coordinate where their bits differ (their XOR is 1). Charlie knows which coordinate is the “overlap” but does not know the actual bits. The goal is for Charlie to output the XOR value. The inputs are correlated, and the presence of ⊥ symbols prevents reductions from classic problems such as set disjointness or indexing.

The authors prove a deterministic lower bound for UniqueOverlap using combinatorial arguments based on cross‑intersecting set families. Specifically, they show that for sufficiently large m there exists a support size s = Θ(m) such that any deterministic one‑way protocol in the simultaneous 3‑party model requires Ω(m) bits of communication. This lower bound is then embedded into the hard graph construction, linking the communication cost of UniqueOverlap to the sketch size needed for k‑edge connectivity.

A crucial innovation is a zero‑error simulation that avoids the “partial simulation” technique used in earlier randomized lower bounds, which omitted the messages of Vᵣ vertices and introduced a small error probability. Instead, the authors argue that any deterministic algorithm must produce sketches that are “compatible” with the limited knowledge the referee has about v_σ’s neighborhood. By carefully analyzing the structure of deterministic protocols, they show that the referee can reconstruct v_σ’s neighbor set exactly without any error, even though the Vᵣ vertices are not explicitly simulated. This yields a fully deterministic reduction from UniqueOverlap to k‑edge connectivity.

The main results are formalized as:

• Theorem 1: Every deterministic algorithm deciding k‑edge connectivity on n‑node graphs in the distributed sketching model requires worst‑case sketch length Ω(k) bits, for any super‑constant k ≤ γ√n.
• Theorem 2: The deterministic simultaneous 3‑party communication complexity of UniqueOverlap₍ₘ,ₛ₎ is Ω(m) bits when the support size s = Θ(m).

These contributions close a longstanding open problem listed in Assadi’s survey (2022) concerning deterministic lower bounds for connectivity in the sketching model. Moreover, the techniques—particularly the UniqueOverlap problem and the zero‑error simulation—provide a new toolkit for proving deterministic lower bounds for other graph problems in distributed sketching, such as minimum cuts, sparsifiers, or fault‑tolerant subgraph constructions. The paper thus advances our understanding of the inherent information requirements for exact graph property testing in a highly constrained distributed setting.