# A Weighted Approach to the Maximum Cardinality Bipartite Matching Problem with Applications in Geometric Settings

## TL;DR

This paper introduces a new weighted algorithm for maximum bipartite matching that improves efficiency, especially in geometric settings, by leveraging edge weights and graph structure to outperform classical algorithms.

## Contribution

The paper presents a novel weighted approach that reduces the time complexity for maximum bipartite matching in specific graph classes, extending to geometric applications and graphs with separators.

## Key findings

- Achieves faster matching algorithms in weighted bipartite graphs with certain structures.
- Provides improved approximate algorithms for geometric bottleneck matching.
- Extends applicability to graphs with balanced vertex separators.

## Abstract

We present a weighted approach to compute a maximum cardinality matching in an arbitrary bipartite graph. Our main result is a new algorithm that takes as input a weighted bipartite graph $G(A\cup B,E)$ with edge weights of $0$ or $1$. Let $w \leq n$ be an upper bound on the weight of any matching in $G$. Consider the subgraph induced by all the edges of $G$ with a weight $0$. Suppose every connected component in this subgraph has $\mathcal{O}(r)$ vertices and $\mathcal{O}(mr/n)$ edges. We present an algorithm to compute a maximum cardinality matching in $G$ in $\tilde{\mathcal{O}}( m(\sqrt{w}+ \sqrt{r}+\frac{wr}{n}))$ time. When all the edge weights are $1$ (symmetrically when all weights are $0$), our algorithm will be identical to the well-known Hopcroft-Karp (HK) algorithm, which runs in $\mathcal{O}(m\sqrt{n})$ time. However, if we can carefully assign weights of $0$ and $1$ on its edges such that both $w$ and $r$ are sub-linear in $n$ and $wr=\mathcal{O}(n^{\gamma})$ for $\gamma < 3/2$, then we can compute maximum cardinality matching in $G$ in $o(m\sqrt{n})$ time. Using our algorithm, we obtain a new $\tilde{\mathcal{O}}(n^{4/3}/\varepsilon^4)$ time algorithm to compute an $\varepsilon$-approximate bottleneck matching of $A,B\subset\mathbb{R}^2$ and an $\frac{1}{\varepsilon^{\mathcal{O}(d)}}n^{1+\frac{d-1}{2d-1}}\mathrm{poly}\log n$ time algorithm for computing $\varepsilon$-approximate bottleneck matching in $d$-dimensions. All previous algorithms take $\Omega(n^{3/2})$ time. Given any graph $G(A \cup B,E)$ that has an easily computable balanced vertex separator for every subgraph $G'(V',E')$ of size $|V'|^{\delta}$, for $\delta\in [1/2,1)$, we can apply our algorithm to compute a maximum matching in $\tilde{\mathcal{O}}(mn^{\frac{\delta}{1+\delta}})$ time improving upon the $\mathcal{O}(m\sqrt{n})$ time taken by the HK-Algorithm.