Distributed Verification and Hardness of Distributed Approximation (1011.3049v3)

Abstract: We study the {\em verification} problem in distributed networks, stated as follows. Let $H$ be a subgraph of a network $G$ where each vertex of $G$ knows which edges incident on it are in $H$. We would like to verify whether $H$ has some properties, e.g., if it is a tree or if it is connected. We would like to perform this verification in a decentralized fashion via a distributed algorithm. The time complexity of verification is measured as the number of rounds of distributed communication. In this paper we initiate a systematic study of distributed verification, and give almost tight lower bounds on the running time of distributed verification algorithms for many fundamental problems such as connectivity, spanning connected subgraph, and $s-t$ cut verification. We then show applications of these results in deriving strong unconditional time lower bounds on the {\em hardness of distributed approximation} for many classical optimization problems including minimum spanning tree, shortest paths, and minimum cut. Many of these results are the first non-trivial lower bounds for both exact and approximate distributed computation and they resolve previous open questions. Moreover, our unconditional lower bound of approximating minimum spanning tree (MST) subsumes and improves upon the previous hardness of approximation bound of Elkin [STOC 2004] as well as the lower bound for (exact) MST computation of Peleg and Rubinovich [FOCS 1999]. Our result implies that there can be no distributed approximation algorithm for MST that is significantly faster than the current exact algorithm, for {\em any} approximation factor. Our lower bound proofs show an interesting connection between communication complexity and distributed computing which turns out to be useful in establishing the time complexity of exact and approximate distributed computation of many problems.

• The paper establishes lower bounds on distributed subgraph verification, showing that spanning tree checks require Ω(√n + D) rounds.
• The study demonstrates the hardness of fast approximations in distributed algorithms by leveraging communication complexity reductions.
• The authors propose near-optimal algorithms that guide efficient deployment in resource-constrained, dynamic networked systems.

Analysis of Distributed Verification and Hardness of Distributed Approximation

This paper conducts a rigorous exploration of distributed verification within networked systems and extends the implications of these findings to approximation challenges in distributed computing. Starting from a clearly defined problem of subgraph verification, it systematically derives lower bounds for various distributed problems such as connectivity, spanning subgraphs, and cut verifications, along with unconditional lower bounds for distributed algorithms involving minimum spanning trees (MST), shortest paths, and minimum cuts.

The authors introduce a structured approach to tackle the distributed verification problem where each node acknowledges its local set of edges forming part of a subgraph $H$, and verification against specific properties (e.g., connectivity or spanning tree status) needs to be validated throughout the network. The primary metric is the number of communication rounds required for a distributed algorithm to perform this verification; thus, the emphasis is on the time complexity in distributed setups.

Central Contributions

1. Lower Bounds on Verification Time: The paper formulates specific lower bounds on the time complexity needed for distributed verification algorithms. These bounds are shown to be nearly tight and apply to a broad spectrum of problems, marking the first substantial progress in setting non-trivial lower boundaries for both exact and approximative distributed computations. For example, verifying if a subgraph is a spanning tree requires $\Omega(\sqrt{n} + D)$ rounds, where $D$ is the network's diameter and $n$ is the number of nodes.
2. Hardness of Approximation in Distributed Contexts: By employing communication complexity lower bounds, the paper establishes groundbreaking connections that lead to strong evidence against the feasibility of fast approximation algorithms for key distributed problems like MST. Specifically, the paper extends earlier MST approximability bounds, refuting faster-than-exact solutions even with relaxed precision constraints.
3. Utilization of Communication Complexity: Via innovative reductions, the paper bridges distributed computing's verification challenges with set-disjointness and equality problems from communication complexity. This ties distributed verification's computational hardness to well-established complexity insights, solidifying the theoretical underpinnings of these distributed obstructions.
4. Optimistic Upper Bounds: Despite the punitive complexity assertions, the paper affirms the existence of algorithms that solve almost all considered verification problems within $O(\sqrt{n} \log n + D)$ rounds, thereby demonstrating the practical reach towards these complexity bounds in distributed environments.
5. Expansive Implications for Networked Systems: Beyond theoretical assertions, these results bear significant weight in practical terms, particularly in resource-constrained and dynamically evolving networks such as sensor or peer-to-peer architectures. Here, even approximate results bear the loudest consequences for efficient distributed processing.

Practical and Theoretical Implications

The implications extend both theoretically and practically. From a theoretical standpoint, the work resolves older open issues, subsumes previous bounds, and inspires new enquiry areas within distributed network verification. Practically, the bounds inform the limitations of distributed systems, crucial for practitioners aiming to optimize algorithm deployment strategies under constrained or volatile network conditions.

Speculation for Future Directions

The profound ties established between distributed and communication complexities open new queries, especially in extending similar reduction techniques to alternate distributed challenges or exploring verification methods in non-traditional network structures, which may lead to weakened bounds or specialized optimization strategies.

In conclusion, this paper makes substantial strides in defining distributed verification's limits and draws inherent connections to approximation within decentralized frameworks, underpinned by communication complexity insights. This thoughtful exposition not only advances distributed computing theory but also guides practical deployment in scalable networked systems.