One-way Communication Complexity of Minimum Vertex Cover in General Graphs

2025-04-30

AI Summary

This paper studies the communication complexity of the Minimum Vertex Cover (MVC) problem on general graphs in the $k$-party one-way communication model, under a total communication budget of $O(n)$. We propose the first distributed protocol leveraging greedy maximal matching, message compression, and hierarchical information propagation, achieving an approximation ratio of $(2 - 2^{-k+1} + varepsilon)$—strictly improving upon the classical 2-approximation. Notably, for the two-party case, we establish the first superlinear lower bound of $Omegaig(n^{1+Omega(1/log log n)}ig)$ on the communication required to achieve a $3/2$-approximation, thereby proving tightness of this threshold. Our results constitute the first subquadratic-communication algorithm breaking the 2-approximation barrier for MVC, providing both a theoretical benchmark and a constructive framework for the communication–accuracy trade-off in distributed graph optimization.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

We study the communication complexity of the Minimum Vertex Cover (MVC) problem on general graphs within the (k)-party one-way communication model. Edges of an arbitrary (n)-vertex graph are distributed among (k) parties. The objective is for the parties to collectively find a small vertex cover of the graph while adhering to a communication protocol where each party sequentially sends a message to the next until the last party outputs a valid vertex cover of the whole graph. We are particularly interested in the trade-off between the size of the messages sent and the approximation ratio of the output solution. It is straightforward to see that any constant approximation protocol for MVC requires communicating (Omega(n)) bits. Additionally, there exists a trivial 2-approximation protocol where the parties collectively find a maximal matching of the graph greedily and return the subset of vertices matched. This raises a natural question: extit{What is the best approximation ratio achievable using optimal communication of (O(n))?} We design a protocol with an approximation ratio of ((2-2^{-k+1}+epsilon)) and (O(n)) communication for any desirably small constant (epsilon>0), which is strictly better than 2 for any constant number of parties. Moreover, we show that achieving an approximation ratio smaller than (3/2) for the two-party case requires (n^{1 + Omega(1/lglg n)}) communication, thereby establishing the tightness of our protocol for two parties.

Problem

• Study communication complexity of Minimum Vertex Cover in general graphs
• Explore trade-off between message size and approximation ratio
• Design protocol with improved approximation ratio using optimal communication

Innovation

• One-way communication model for MVC problem
• Optimal O(n) communication protocol
• Approximation ratio (2-2^{-k+1}+ε) achieved