R(QPS-Serena) and R(QPS-Serenade): Two Novel Augmenting-Path Based Algorithms for Computing Approximate Maximum Weight Matching

TL;DR

This paper introduces two new algorithms, R(QPS-Serena) and R(QPS-Serenade), that efficiently approximate maximum weight matching in bipartite graphs, matching state-of-the-art performance empirically but lacking theoretical guarantees.

Contribution

The paper presents novel augmenting-path based algorithms that convert existing algorithms into approximate maximum weight matching algorithms with promising empirical efficiency.

Findings

R(QPS-Serena) computes (1-ε)-MWM in linear time for certain graphs.

R(QPS-Serenade) computes (1-ε)-MWM in O(N log N) time empirically.

Both algorithms match the performance of current best solutions empirically.

Abstract

In this addendum, we show that the switching algorithm QPS-SERENA can be converted R(QPS-SERENA), an algorithm for computing approximate Maximum Weight Matching (MWM). Empirically, R(QPS-SERENA) computes $(1-\epsilon)$-MWM within linear time (with respect to the number of edges $N^2$) for any fixed $\epsilon\in (0,1)$, for complete bipartite graphs with {\it i.i.d.} uniform edge weight distributions. This efficacy matches that of the state-of-art solution, although we so far cannot prove any theoretical guarantees on the time complexities needed to attain a certain approximation ratio. Then, we have similarly converted QPS-SERENADE to R(QPS-SERENADE), which empirically should output $(1-\epsilon)$-MWM within only $O(N \log N)$ time for the same type of complete bipartite graphs as described above.