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

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