# A 2/3-Approximation Algorithm for Vertex-weighted Matching

## TL;DR

This paper presents a 2/3-approximation algorithm for the maximum vertex-weighted matching problem in non-bipartite graphs, with efficient implementation and superior performance on large graphs compared to existing algorithms.

## Contribution

It introduces a novel 2/3-approximation algorithm for non-bipartite graphs by limiting augmenting path length, with proven approximation ratio and practical efficiency.

## Key findings

- Algorithm runs in under a minute on large graphs with tens of millions of vertices.
- Outperforms existing algorithms in speed and solution quality on test problems.
- Handles graphs where exact algorithms fail to terminate within 100 hours.

## Abstract

We consider the maximum vertex-weighted matching problem (MVM) for non-bipartite graphs. In earlier work we have described a 2/3-approximation algorithm for the MVM on bipartite graphs (Dobrian, Halappanavar, Pothen and Al-Herz, SIAM J. Scientific Computing, 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 $O(m \log \Delta + n \log n)$, where $n$ is the number of vertices, $m$ is the number of edges, and $\Delta$ 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. In our test set of nineteen problems, there are graphs on which the exact algorithms fail to terminate in 100 hours. 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.