A Batch-dynamic Suitor Algorithm for Approximating Maximum Weighted Matching

Abstract

Matching is a popular combinatorial optimization problem with numerous applications in both commercial and scientific fields. Computing optimal matchings w.r.t. cardinality or weight can be done in polynomial time; still, this task can become infeasible for very large networks. Thus, several approximation algorithms that trade solution quality for a faster running time have been proposed. For networks that change over time, fully dynamic algorithms that efficiently maintain an approximation of the optimal matching after a graph update have been introduced as well. However, no semi- or fully dynamic algorithm for (approximate) maximum weighted matching has been implemented. In this article, we focus on the problem of maintaining a \( 1/2 \) -approximation of a maximum weighted matching (MWM) in fully dynamic graphs. Limitations of existing algorithms for this problem are (i) high constant factors in their time complexity, (ii) the fact that none of them supports batch updates, and (iii) the lack of a practical implementation, meaning that their actual performance on real-world graphs has not been investigated. We propose and implement a new batch-dynamic \( 1/2 \) -approximation algorithm for MWM based on the Suitor algorithm and its local edge domination strategy [Manne and Halappanavar, IPDPS 2014]. We provide a detailed analysis of our algorithm and prove its approximation guarantee. Despite having a worst-case running time of \( \mathcal {O}(n + m) \) for a single graph update, our extensive experimental evaluation shows that our algorithm is much faster in practice. For example, compared to a static recomputation with sequential Suitor , single-edge updates are handled up to \( 10^5\times \) to \( 10^6\times \) faster, while batches of \( 10^4 \) edge updates are handled up to \( 10^2\times \) to \( 10^3\times \) faster.