A 2/3-approximation algorithm for vertex-weighted matching

Abstract

We consider the maximum vertex-weighted matching problem (MVM) for non-bipartite graphs in which non-negative weights are assigned to the vertices of a graph and a matching that maximizes the sum of the weights of the matched vertices is desired. In earlier work we have described a 2/3-approximation algorithm for the MVM on bipartite graphs (Florin Dobrian et al., 2019). Here we show that a 2/3-approximation algorithm for MVM on non-bipartite graphs can be obtained by restricting the length of augmenting paths to at most three. The algorithm has time complexity O(mlogΔ+nlogn), where n is the number of vertices, m is the number of edges, and Δ is the maximum degree of a vertex.The approximation ratio of the algorithm is obtained by considering failed vertices, i.e., vertices that the approximation algorithm fails to match but the exact algorithm does. We show that there are two distinct heavier matched vertices that we can charge each failed vertex to. Our proof techniques characterize the structure of augmenting paths in a novel way.We have implemented the 2/3-approximation algorithm and show that it runs in under a minute on graphs with tens of millions of vertices and hundreds of millions of edges. We compare its performance with five other algorithms: an exact algorithm for MVM, an exact algorithm for the maximum edge-weighted matching (MEM) problem, as well as three approximation algorithms. The approximation algorithms include a 1/2-approximation algorithm for MVM, and (2/3−ε)- and (1−ε)-approximation algorithms for the MEM. In our test set of nineteen problems, there are graphs on which the exact algorithms fail to terminate in 100 hours. In addition, the new 2/3-approximation algorithm for MVM outperforms the other approximation algorithms by either being faster (often by orders of magnitude) or obtaining better weights.