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Let $G = (X+Y, E)$ be a bipartite graph, and suppose we want to find a maximum-cardinality matching in $G$. The Hopcroft-Karp algorithm runs in time $O(|E|\sqrt{|V|})$, where here $|V| = |X|+|Y|$. So if the graph is unbalanced and $|X|<|Y|$, the run-time is $O(|E|\sqrt{|Y|})$.

Is there an algorithm that attains a better worst-case runtime complexity for the case of an unbalanced bipartite graph, where $|X| \in o(|Y|)$? In particular, if $|X|$ is constant, is it possible to get run-time $O(|E|)$?

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I just found the answer myself. In this paper:

Lyle Ramshaw, Robert E. Tarjan (2012). "On minimum-cost assignments in unbalanced bipartite graphs‏". Technical reports, HP research labs.

in section 5, the authors show that the Hopcroft-Karp algorithm in fact solves the following problem: given an integer $s$, find matchings with $1,\ldots,s$ edges. The run-time is $O(|E|\sqrt{s})$. In particular, if $|X|<|Y|$, we can take $s=|X|$ and the run-time is $O(|E|\sqrt{|X|})$.

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    $\begingroup$ I would suggest this link arxiv.org/abs/1904.11244 as an additional note on parameterized matching. It proves square root dependence on many parameters, i.e. $O(|E|\sqrt k)$, for different parameters $k$. $\endgroup$ – narek Bojikian Sep 16 at 0:40

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