Maximum-cardinality matching in unbalanced bipartite graphsExistence of bipartite perfect matchingmaximum bipartite matchingVertex cover in bipartite graph from Hopcroft-Karp AlgorithmMaximum bipartite matching when some nodes must be matchedWhy is bipartite graph matching hard?Understanding characterizations of Matching on GraphsMore efficient maximum bipartite matching
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Maximum-cardinality matching in unbalanced bipartite graphs
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Maximum-cardinality matching in unbalanced bipartite graphs
Existence of bipartite perfect matchingmaximum bipartite matchingVertex cover in bipartite graph from Hopcroft-Karp AlgorithmMaximum bipartite matching when some nodes must be matchedWhy is bipartite graph matching hard?Understanding characterizations of Matching on GraphsMore efficient maximum bipartite matching
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$begingroup$
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 $O(|E|sqrt)$, where here $|V| = |X|+|Y|$. So if the graph is unbalanced and $|X|<|Y|$, the run-time is $O(|E|sqrt)$.
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 $O(|E|)$?
optimization reference-request runtime-analysis bipartite-matching
asked 8 hours ago
Erel Segal-HaleviErel Segal-Halevi
2,60713 silver badges44 bronze badges
$endgroup$
add a comment
|
$begingroup$
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 $O(|E|sqrt)$, where here $|V| = |X|+|Y|$. So if the graph is unbalanced and $|X|<|Y|$, the run-time is $O(|E|sqrt)$.
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 $O(|E|)$?
optimization reference-request runtime-analysis bipartite-matching
asked 8 hours ago
Erel Segal-HaleviErel Segal-Halevi
2,60713 silver badges44 bronze badges
$endgroup$
add a comment
|
$begingroup$
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 $O(|E|sqrt)$, where here $|V| = |X|+|Y|$. So if the graph is unbalanced and $|X|<|Y|$, the run-time is $O(|E|sqrt)$.
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 $O(|E|)$?
optimization reference-request runtime-analysis bipartite-matching
asked 8 hours ago
Erel Segal-HaleviErel Segal-Halevi
2,60713 silver badges44 bronze badges
$endgroup$
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 $O(|E|sqrt)$, where here $|V| = |X|+|Y|$. So if the graph is unbalanced and $|X|<|Y|$, the run-time is $O(|E|sqrt)$.
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 $O(|E|)$?
optimization reference-request runtime-analysis bipartite-matching
optimization reference-request runtime-analysis bipartite-matching
asked 8 hours ago
Erel Segal-HaleviErel Segal-Halevi
2,60713 silver badges44 bronze badges
asked 8 hours ago
Erel Segal-HaleviErel Segal-Halevi
2,60713 silver badges44 bronze badges
asked 8 hours ago
Erel Segal-HaleviErel Segal-Halevi
2,60713 silver badges44 bronze badges
asked 8 hours ago
Erel Segal-HaleviErel Segal-Halevi
2,60713 silver badges44 bronze badges
asked 8 hours ago
Erel Segal-HaleviErel Segal-Halevi
2,60713 silver badges44 bronze badges
2,60713 silver badges44 bronze badges
add a comment
|
add a comment
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1 Answer
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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 $O(|E|sqrts)$. In particular, if $|X|<|Y|$, we can take $s=|X|$ and the run-time is $O(|E|sqrtX)$.
answered 7 hours ago
Erel Segal-HaleviErel Segal-Halevi
2,60713 silver badges44 bronze badges
$endgroup$
$begingroup$
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
2 hours ago
add a comment
|
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1 Answer
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1 Answer
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$begingroup$
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 $O(|E|sqrts)$. In particular, if $|X|<|Y|$, we can take $s=|X|$ and the run-time is $O(|E|sqrtX)$.
answered 7 hours ago
Erel Segal-HaleviErel Segal-Halevi
2,60713 silver badges44 bronze badges
$endgroup$
$begingroup$
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
2 hours ago
add a comment
|
$begingroup$
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 $O(|E|sqrts)$. In particular, if $|X|<|Y|$, we can take $s=|X|$ and the run-time is $O(|E|sqrtX)$.
answered 7 hours ago
Erel Segal-HaleviErel Segal-Halevi
2,60713 silver badges44 bronze badges
$endgroup$
$begingroup$
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
2 hours ago
add a comment
|
$begingroup$
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 $O(|E|sqrts)$. In particular, if $|X|<|Y|$, we can take $s=|X|$ and the run-time is $O(|E|sqrtX)$.
answered 7 hours ago
Erel Segal-HaleviErel Segal-Halevi
2,60713 silver badges44 bronze badges
$endgroup$
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 $O(|E|sqrts)$. In particular, if $|X|<|Y|$, we can take $s=|X|$ and the run-time is $O(|E|sqrtX)$.
answered 7 hours ago
Erel Segal-HaleviErel Segal-Halevi
2,60713 silver badges44 bronze badges
answered 7 hours ago
Erel Segal-HaleviErel Segal-Halevi
2,60713 silver badges44 bronze badges
answered 7 hours ago
Erel Segal-HaleviErel Segal-Halevi
2,60713 silver badges44 bronze badges
answered 7 hours ago
Erel Segal-HaleviErel Segal-Halevi
2,60713 silver badges44 bronze badges
2,60713 silver badges44 bronze badges
$begingroup$
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
2 hours ago
add a comment
|
$begingroup$
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
2 hours ago
$begingroup$
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
2 hours ago
$begingroup$
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
2 hours ago
add a comment
|
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