ℓp-Norm Multiway Cut
We introduce and study ℓ p - norm - multiway - cut : the input here is an undirected graph with non-negative edge weights along with k terminals and the goal is to find a partition of the vertex set into k parts each containing exactly one terminal so as to minimize the ℓ p -norm of the cut values o...
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| Veröffentlicht in: | Algorithmica 2022, Vol.84 (9), p.2667-2701 |
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| container_title | Algorithmica |
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| creator | Chandrasekaran, Karthekeyan Wang, Weihang |
| description | We introduce and study
ℓ
p
-
norm
-
multiway
-
cut
: the input here is an undirected graph with non-negative edge weights along with
k
terminals and the goal is to find a partition of the vertex set into
k
parts each containing exactly one terminal so as to minimize the
ℓ
p
-norm of the cut values of the parts. This is a unified generalization of min-sum multiway cut (when
p
=
1
) and min–max multiway cut (when
p
=
∞
), both of which are well-studied classic problems in the graph partitioning literature. We show that
ℓ
p
-
norm
-
multiway
-
cut
is NP-hard for constant number of terminals and is NP-hard in planar graphs. On the algorithmic side, we design an
O
(
log
1.5
n
log
0.5
k
)
-approximation for all
p
≥
1
. We also show an integrality gap of
Ω
(
k
1
-
1
/
p
)
for a natural convex program and an
O
(
k
1
-
1
/
p
-
ϵ
)
-inapproximability for any constant
ϵ
>
0
assuming the small set expansion hypothesis. |
| doi_str_mv | 10.1007/s00453-022-00983-3 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_sprin</sourceid><recordid>TN_cdi_proquest_journals_2701552402</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2701552402</sourcerecordid><originalsourceid>FETCH-LOGICAL-p1423-65c74296b03d5eae16801caf46b4cc601749d0fe7877be0ff365e584009562e83</originalsourceid><addsrcrecordid>eNpFkMtKxEAQRRtRMI5uXbgacN1a1VX9yFKCLxh1o-smyXRkhnES0wni3j_wD_0SWyO4uhQc6l6OECcIZwhgzyMAa5KglATIHUnaERkypVMz7ooM0DrJBu2-OIhxDYDK5iYTx18fn528b_uX-d24GVZv5fu8GIdDsdeUmxiO_nImnq4uH4sbuXi4vi0uFrJDViSNri2r3FRASx3KgMYB1mXDpuK6NqmU8yU0wTprqwBNQ0YH7Tht1EYFRzNxOv3t-vZ1DHHw63bst6nSKwuotWJQiaKJil2_2j6H_p9C8D8C_CTAJwH-V4An-gZr2Etj</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2701552402</pqid></control><display><type>article</type><title>ℓp-Norm Multiway Cut</title><source>SpringerLink Journals - AutoHoldings</source><creator>Chandrasekaran, Karthekeyan ; Wang, Weihang</creator><creatorcontrib>Chandrasekaran, Karthekeyan ; Wang, Weihang</creatorcontrib><description>We introduce and study
ℓ
p
-
norm
-
multiway
-
cut
: the input here is an undirected graph with non-negative edge weights along with
k
terminals and the goal is to find a partition of the vertex set into
k
parts each containing exactly one terminal so as to minimize the
ℓ
p
-norm of the cut values of the parts. This is a unified generalization of min-sum multiway cut (when
p
=
1
) and min–max multiway cut (when
p
=
∞
), both of which are well-studied classic problems in the graph partitioning literature. We show that
ℓ
p
-
norm
-
multiway
-
cut
is NP-hard for constant number of terminals and is NP-hard in planar graphs. On the algorithmic side, we design an
O
(
log
1.5
n
log
0.5
k
)
-approximation for all
p
≥
1
. We also show an integrality gap of
Ω
(
k
1
-
1
/
p
)
for a natural convex program and an
O
(
k
1
-
1
/
p
-
ϵ
)
-inapproximability for any constant
ϵ
>
0
assuming the small set expansion hypothesis.</description><identifier>ISSN: 0178-4617</identifier><identifier>EISSN: 1432-0541</identifier><identifier>DOI: 10.1007/s00453-022-00983-3</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Algorithm Analysis and Problem Complexity ; Algorithms ; Computer Science ; Computer Systems Organization and Communication Networks ; Data Structures and Information Theory ; Mathematics of Computing ; Terminals ; Theory of Computation ; Vertex sets</subject><ispartof>Algorithmica, 2022, Vol.84 (9), p.2667-2701</ispartof><rights>The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022</rights><rights>The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><orcidid>0000-0002-0628-5532</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00453-022-00983-3$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00453-022-00983-3$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Chandrasekaran, Karthekeyan</creatorcontrib><creatorcontrib>Wang, Weihang</creatorcontrib><title>ℓp-Norm Multiway Cut</title><title>Algorithmica</title><addtitle>Algorithmica</addtitle><description>We introduce and study
ℓ
p
-
norm
-
multiway
-
cut
: the input here is an undirected graph with non-negative edge weights along with
k
terminals and the goal is to find a partition of the vertex set into
k
parts each containing exactly one terminal so as to minimize the
ℓ
p
-norm of the cut values of the parts. This is a unified generalization of min-sum multiway cut (when
p
=
1
) and min–max multiway cut (when
p
=
∞
), both of which are well-studied classic problems in the graph partitioning literature. We show that
ℓ
p
-
norm
-
multiway
-
cut
is NP-hard for constant number of terminals and is NP-hard in planar graphs. On the algorithmic side, we design an
O
(
log
1.5
n
log
0.5
k
)
-approximation for all
p
≥
1
. We also show an integrality gap of
Ω
(
k
1
-
1
/
p
)
for a natural convex program and an
O
(
k
1
-
1
/
p
-
ϵ
)
-inapproximability for any constant
ϵ
>
0
assuming the small set expansion hypothesis.</description><subject>Algorithm Analysis and Problem Complexity</subject><subject>Algorithms</subject><subject>Computer Science</subject><subject>Computer Systems Organization and Communication Networks</subject><subject>Data Structures and Information Theory</subject><subject>Mathematics of Computing</subject><subject>Terminals</subject><subject>Theory of Computation</subject><subject>Vertex sets</subject><issn>0178-4617</issn><issn>1432-0541</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2022</creationdate><recordtype>article</recordtype><sourceid/><recordid>eNpFkMtKxEAQRRtRMI5uXbgacN1a1VX9yFKCLxh1o-smyXRkhnES0wni3j_wD_0SWyO4uhQc6l6OECcIZwhgzyMAa5KglATIHUnaERkypVMz7ooM0DrJBu2-OIhxDYDK5iYTx18fn528b_uX-d24GVZv5fu8GIdDsdeUmxiO_nImnq4uH4sbuXi4vi0uFrJDViSNri2r3FRASx3KgMYB1mXDpuK6NqmU8yU0wTprqwBNQ0YH7Tht1EYFRzNxOv3t-vZ1DHHw63bst6nSKwuotWJQiaKJil2_2j6H_p9C8D8C_CTAJwH-V4An-gZr2Etj</recordid><startdate>2022</startdate><enddate>2022</enddate><creator>Chandrasekaran, Karthekeyan</creator><creator>Wang, Weihang</creator><general>Springer US</general><general>Springer Nature B.V</general><scope/><orcidid>https://orcid.org/0000-0002-0628-5532</orcidid></search><sort><creationdate>2022</creationdate><title>ℓp-Norm Multiway Cut</title><author>Chandrasekaran, Karthekeyan ; Wang, Weihang</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-p1423-65c74296b03d5eae16801caf46b4cc601749d0fe7877be0ff365e584009562e83</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2022</creationdate><topic>Algorithm Analysis and Problem Complexity</topic><topic>Algorithms</topic><topic>Computer Science</topic><topic>Computer Systems Organization and Communication Networks</topic><topic>Data Structures and Information Theory</topic><topic>Mathematics of Computing</topic><topic>Terminals</topic><topic>Theory of Computation</topic><topic>Vertex sets</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Chandrasekaran, Karthekeyan</creatorcontrib><creatorcontrib>Wang, Weihang</creatorcontrib><jtitle>Algorithmica</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Chandrasekaran, Karthekeyan</au><au>Wang, Weihang</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>ℓp-Norm Multiway Cut</atitle><jtitle>Algorithmica</jtitle><stitle>Algorithmica</stitle><date>2022</date><risdate>2022</risdate><volume>84</volume><issue>9</issue><spage>2667</spage><epage>2701</epage><pages>2667-2701</pages><issn>0178-4617</issn><eissn>1432-0541</eissn><abstract>We introduce and study
ℓ
p
-
norm
-
multiway
-
cut
: the input here is an undirected graph with non-negative edge weights along with
k
terminals and the goal is to find a partition of the vertex set into
k
parts each containing exactly one terminal so as to minimize the
ℓ
p
-norm of the cut values of the parts. This is a unified generalization of min-sum multiway cut (when
p
=
1
) and min–max multiway cut (when
p
=
∞
), both of which are well-studied classic problems in the graph partitioning literature. We show that
ℓ
p
-
norm
-
multiway
-
cut
is NP-hard for constant number of terminals and is NP-hard in planar graphs. On the algorithmic side, we design an
O
(
log
1.5
n
log
0.5
k
)
-approximation for all
p
≥
1
. We also show an integrality gap of
Ω
(
k
1
-
1
/
p
)
for a natural convex program and an
O
(
k
1
-
1
/
p
-
ϵ
)
-inapproximability for any constant
ϵ
>
0
assuming the small set expansion hypothesis.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s00453-022-00983-3</doi><tpages>35</tpages><orcidid>https://orcid.org/0000-0002-0628-5532</orcidid></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0178-4617 |
| ispartof | Algorithmica, 2022, Vol.84 (9), p.2667-2701 |
| issn | 0178-4617 1432-0541 |
| language | eng |
| recordid | cdi_proquest_journals_2701552402 |
| source | SpringerLink Journals - AutoHoldings |
| subjects | Algorithm Analysis and Problem Complexity Algorithms Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Mathematics of Computing Terminals Theory of Computation Vertex sets |
| title | ℓp-Norm Multiway Cut |
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