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

TL;DR

This work analyzes MVC in the k-party one-way communication model for general graphs, establishing a protocol that achieves an expected approximation of $(2 - 2^{-k+1} + ε)$ with $O(n)$ communication, for any constant ε>0. The authors introduce a weight-based, multi-party construction that communicates only vertex-cover sketches rather than edges, and they prove a key two-player MVC game bound that feeds into an inductive k-party protocol with a tight two-party lower bound. The results show that beating 2-approximation is possible with linear communication for constant k, and that the two-party threshold of $3/2$ is essentially tight up to superlinear communication, highlighting a clear view of how the approximation-communication trade-off scales with the number of parties. The work also connects to streaming lower bounds and suggests potential avenues for space-efficient streaming algorithms, by showing the insufficiency of constant-party one-way bounds in ruling out better-than-2 MVC in streaming settings.

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 \(Ω(n)\) 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: \textit{What is the best approximation ratio achievable using optimal communication of \(O(n)\)?} We design a protocol with an approximation ratio of \((2-2^{-k+1}+ε)\) and \(O(n)\) communication for any desirably small constant \(ε>0\), 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 \(n^{1 + Ω(1/\lg\lg n)}\) communication, thereby establishing the tightness of our protocol for two parties.

Figures (2)

• Figure 1: The blue matching $M_1$ is given to the first party, the red matching $M_2$ to the second party, and a complete graph (non-bipartite) among vertices in the box to the third party.
• Figure 2: The tree of communications generated by parties using algorithm \ref{['algk']}. The message $S$ sent through each edge to the $i$-th layer is a vertex cover of the subgraphs given to the first $i-1$ parties. For any of these messages the $i$-th party receives, she constructs $b_i$ different vertex covers of $G'_i=G_i[V\setminus S]$ and sends their union with $S$ to the next party. Each edge basically represents a subproblem party $i$ needs to solve. Given $S$, she will make $b_i$ guesses about the size of $\tau\left(G'_i \right)$ and for any of these guesses, needs to solve the following subproblem: Find a vertex cover $X$ of $G'_i$ such that committing the subset $S\cup X$ to the final solution results in a good approximation ratio conditioned on the specific guess about $\tau\left(G'_i \right)$.

Theorems & Definitions (20)

• Theorem 1.1
• Proposition 2.1
• Definition 3.1: MVC Game
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Lemma 3.5