A weighted approach to the maximum cardinality bipartite matching problem with applications in geometric settings

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 $O(r)$ vertices and $O(mr/n)$ edges. We present an algorithm to compute a maximum cardinality matching in $G$ in $\tilde{O}( m(\sqrt{w}+ \sqrt{r}+\frac{wr}{n}))$ 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 $O(m\sqrt{n})$ 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 $wr=O(n^{\gamma})$ for $\gamma < 3/2$, then we can compute a maximum cardinality matching in $o(m\sqrt{n})$ time. Using our algorithm, we obtain a new $\tilde{O}(n^{4/3}/\varepsilon^3)$ time algorithm to compute an $\varepsilon$-approximate bottleneck matching of $A,B\subset \mathbb{R}^2$ and a $\frac{1}{\varepsilon^{O(d)}}n^{1+\frac{d-1}{2d-1}}\mathrm{poly}\log n$ time algorithm for computing an $\varepsilon$-approximate bottleneck matching in $d$-dimensions. All previous algorithms take $\Omega(n^{3/2})$ time.Our algorithm also applies to any graph $G(A \cup B,E)$ that has an easily computable balanced vertex separator of size $|V'|^{\delta}$, for every subgraph $G'(V',E')$ where $\delta\in [1/2,1)$. By applying our algorithm, we can compute a maximum matching in $\tilde{O}(mn^{\frac{\delta}{1+\delta}})$ time, improving upon the $O(m\sqrt{n})$ time taken by the HK-Algorithm.