Strongly stable matchings in time O ( nm ) and extension to the hospitals-residents problem
An instance of the stable marriage problem is an undirected bipartite graph G = ( X ∪ W , E ) with linearly ordered adjacency lists with ties allowed in the ordering. A matching M is a set of edges, no two of which share an endpoint. An edge e = ( a , b ) ∈ E ∖ M is a blocking edge for M if a is eit...
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| Veröffentlicht in: | ACM transactions on algorithms 2007-05, Vol.3 (2), p.15 |
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| creator | Kavitha, Telikepalli Mehlhorn, Kurt Michail, Dimitrios Paluch, Katarzyna E. |
| description | An instance of the stable marriage problem is an undirected bipartite graph
G
= (
X
∪
W
,
E
) with linearly ordered adjacency lists with ties allowed in the ordering. A matching
M
is a set of edges, no two of which share an endpoint. An edge
e
= (
a
,
b
) ∈
E
∖
M
is a
blocking edge
for
M
if
a
is either unmatched or strictly prefers
b
to its partner in
M
, and
b
is unmatched, strictly prefers
a
to its partner in
M
, or is indifferent between them. A matching is strongly stable if there is no blocking edge with respect to it. We give an
O
(
nm
) algorithm for computing strongly stable matchings, where
n
is the number of vertices and
m
the number of edges. The previous best algorithm had running time
O
(
m
2
). We also study this problem in the hospitals-residents setting, which is a many-to-one extension of the aforementioned problem. We give an
O
(
m
∑
h∈H
p
h
) algorithm for computing a strongly stable matching in the hospitals-residents problem, where
p
h
is the quota of a hospital
h
. The previous best algorithm had running time
O
(
m
2
). |
| doi_str_mv | 10.1145/1240233.1240238 |
| format | Article |
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G
= (
X
∪
W
,
E
) with linearly ordered adjacency lists with ties allowed in the ordering. A matching
M
is a set of edges, no two of which share an endpoint. An edge
e
= (
a
,
b
) ∈
E
∖
M
is a
blocking edge
for
M
if
a
is either unmatched or strictly prefers
b
to its partner in
M
, and
b
is unmatched, strictly prefers
a
to its partner in
M
, or is indifferent between them. A matching is strongly stable if there is no blocking edge with respect to it. We give an
O
(
nm
) algorithm for computing strongly stable matchings, where
n
is the number of vertices and
m
the number of edges. The previous best algorithm had running time
O
(
m
2
). We also study this problem in the hospitals-residents setting, which is a many-to-one extension of the aforementioned problem. We give an
O
(
m
∑
h∈H
p
h
) algorithm for computing a strongly stable matching in the hospitals-residents problem, where
p
h
is the quota of a hospital
h
. The previous best algorithm had running time
O
(
m
2
).</description><identifier>ISSN: 1549-6325</identifier><identifier>EISSN: 1549-6333</identifier><identifier>DOI: 10.1145/1240233.1240238</identifier><language>eng</language><ispartof>ACM transactions on algorithms, 2007-05, Vol.3 (2), p.15</ispartof><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c272t-c85b14221756e4da2a4eccb8795502f6d1fb787c773a7d4e1770fc8f67a4c7e73</citedby><cites>FETCH-LOGICAL-c272t-c85b14221756e4da2a4eccb8795502f6d1fb787c773a7d4e1770fc8f67a4c7e73</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,780,784,27924,27925</link.rule.ids></links><search><creatorcontrib>Kavitha, Telikepalli</creatorcontrib><creatorcontrib>Mehlhorn, Kurt</creatorcontrib><creatorcontrib>Michail, Dimitrios</creatorcontrib><creatorcontrib>Paluch, Katarzyna E.</creatorcontrib><title>Strongly stable matchings in time O ( nm ) and extension to the hospitals-residents problem</title><title>ACM transactions on algorithms</title><description>An instance of the stable marriage problem is an undirected bipartite graph
G
= (
X
∪
W
,
E
) with linearly ordered adjacency lists with ties allowed in the ordering. A matching
M
is a set of edges, no two of which share an endpoint. An edge
e
= (
a
,
b
) ∈
E
∖
M
is a
blocking edge
for
M
if
a
is either unmatched or strictly prefers
b
to its partner in
M
, and
b
is unmatched, strictly prefers
a
to its partner in
M
, or is indifferent between them. A matching is strongly stable if there is no blocking edge with respect to it. We give an
O
(
nm
) algorithm for computing strongly stable matchings, where
n
is the number of vertices and
m
the number of edges. The previous best algorithm had running time
O
(
m
2
). We also study this problem in the hospitals-residents setting, which is a many-to-one extension of the aforementioned problem. We give an
O
(
m
∑
h∈H
p
h
) algorithm for computing a strongly stable matching in the hospitals-residents problem, where
p
h
is the quota of a hospital
h
. The previous best algorithm had running time
O
(
m
2
).</description><issn>1549-6325</issn><issn>1549-6333</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2007</creationdate><recordtype>article</recordtype><recordid>eNo9kE1LAzEYhIMoWKtnrzmJHrbN5yZ7lGJVKPSgnjws2ey7bWQ3qUkK9t-70uLpGZhhBgahW0pmlAo5p0wQxvnsSH2GJlSKqig55-f_mslLdJXSFyG84lxP0OdbjsFv-gNO2TQ94MFku3V-k7DzOLsB8BrfYz_gB2x8i-Eng08ujF7AeQt4G9LOZdOnIkJyLfic8C6GsWq4RhfdaMDNiVP0sXx6X7wUq_Xz6-JxVVimWC6slg0VjFElSxCtYUaAtY1WlZSEdWVLu0ZpZZXiRrUCqFKks7orlRFWgeJTdHfsHXe_95ByPbhkoe-Nh7BPNSdVxRSXY3B-DNoYUorQ1bvoBhMPNSX134n16cQTNf8FL4xkIg</recordid><startdate>200705</startdate><enddate>200705</enddate><creator>Kavitha, Telikepalli</creator><creator>Mehlhorn, Kurt</creator><creator>Michail, Dimitrios</creator><creator>Paluch, Katarzyna E.</creator><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>8FD</scope><scope>JQ2</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>200705</creationdate><title>Strongly stable matchings in time O ( nm ) and extension to the hospitals-residents problem</title><author>Kavitha, Telikepalli ; Mehlhorn, Kurt ; Michail, Dimitrios ; Paluch, Katarzyna E.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c272t-c85b14221756e4da2a4eccb8795502f6d1fb787c773a7d4e1770fc8f67a4c7e73</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2007</creationdate><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Kavitha, Telikepalli</creatorcontrib><creatorcontrib>Mehlhorn, Kurt</creatorcontrib><creatorcontrib>Michail, Dimitrios</creatorcontrib><creatorcontrib>Paluch, Katarzyna E.</creatorcontrib><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Technology Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>ACM transactions on algorithms</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Kavitha, Telikepalli</au><au>Mehlhorn, Kurt</au><au>Michail, Dimitrios</au><au>Paluch, Katarzyna E.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Strongly stable matchings in time O ( nm ) and extension to the hospitals-residents problem</atitle><jtitle>ACM transactions on algorithms</jtitle><date>2007-05</date><risdate>2007</risdate><volume>3</volume><issue>2</issue><spage>15</spage><pages>15-</pages><issn>1549-6325</issn><eissn>1549-6333</eissn><abstract>An instance of the stable marriage problem is an undirected bipartite graph
G
= (
X
∪
W
,
E
) with linearly ordered adjacency lists with ties allowed in the ordering. A matching
M
is a set of edges, no two of which share an endpoint. An edge
e
= (
a
,
b
) ∈
E
∖
M
is a
blocking edge
for
M
if
a
is either unmatched or strictly prefers
b
to its partner in
M
, and
b
is unmatched, strictly prefers
a
to its partner in
M
, or is indifferent between them. A matching is strongly stable if there is no blocking edge with respect to it. We give an
O
(
nm
) algorithm for computing strongly stable matchings, where
n
is the number of vertices and
m
the number of edges. The previous best algorithm had running time
O
(
m
2
). We also study this problem in the hospitals-residents setting, which is a many-to-one extension of the aforementioned problem. We give an
O
(
m
∑
h∈H
p
h
) algorithm for computing a strongly stable matching in the hospitals-residents problem, where
p
h
is the quota of a hospital
h
. The previous best algorithm had running time
O
(
m
2
).</abstract><doi>10.1145/1240233.1240238</doi></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 1549-6325 |
| ispartof | ACM transactions on algorithms, 2007-05, Vol.3 (2), p.15 |
| issn | 1549-6325 1549-6333 |
| language | eng |
| recordid | cdi_proquest_miscellaneous_30992735 |
| source | ACM Digital Library Complete |
| title | Strongly stable matchings in time O ( nm ) and extension to the hospitals-residents problem |
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