Layers and matroids for the traveling salesman’s paths

Gottschalk and Vygen proved that every solution of the subtour elimination linear program for traveling salesman paths is a convex combination of more and more restrictive “generalized Gao-trees”. We give a short proof of this fact, as a layered convex combination of bases of a sequence of increasin...

Full description

Saved in:
Bibliographic Details
Published in:Operations research letters Vol. 46; no. 1; pp. 60 - 63
Main Authors: Schalekamp, Frans, Sebő, András, Traub, Vera, van Zuylen, Anke
Format: Journal Article
Language:English
Published: Elsevier B.V 01.01.2018
Elsevier
Subjects:
Approximation algorithm
Christofides’ heuristic
Combinatorics
Computer Science
Discrete Mathematics
Mathematics
Matroid partition
Path traveling salesman problem (TSP)
Polyhedra
Spanning tree
Christofides’ heuristic
Polyhedra
Matroid partition
Approximation algorithm
Path traveling salesman problem (TSP)
Spanning tree
ISSN:0167-6377, 1872-7468
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Abstract Gottschalk and Vygen proved that every solution of the subtour elimination linear program for traveling salesman paths is a convex combination of more and more restrictive “generalized Gao-trees”. We give a short proof of this fact, as a layered convex combination of bases of a sequence of increasingly restrictive matroids. A strongly polynomial, combinatorial algorithm follows for finding this convex combination, which is a new tool offering polyhedral insight, already instrumental in recent results for the s−t path TSP.
AbstractList Gottschalk and Vygen proved that every solution of the well-known subtour elimination linear program for traveling salesman paths is a convex combination of a set of more and more restrictive "generalized Gao trees" of the underlying graph. In this paper we give a short proof of this, as a {\em layered} convex combination of bases of a sequence of more and more restrictive matroids. Our proof implies (via the matroid partition theorem) a strongly-polynomial combinatorial algorithm for finding this convex combination. This is a new connection of the TSP to matroids, offering also a new polyhedral insight.
Gottschalk and Vygen proved that every solution of the subtour elimination linear program for traveling salesman paths is a convex combination of more and more restrictive “generalized Gao-trees”. We give a short proof of this fact, as a layered convex combination of bases of a sequence of increasingly restrictive matroids. A strongly polynomial, combinatorial algorithm follows for finding this convex combination, which is a new tool offering polyhedral insight, already instrumental in recent results for the s−t path TSP.
Author Schalekamp, Frans
Sebő, András
Traub, Vera
van Zuylen, Anke
Author_xml – sequence: 1
  givenname: Frans
  surname: Schalekamp
  fullname: Schalekamp, Frans
  email: fms9@cornell.edu
  organization: School of Operations Research and Information Engineering, Department of Computer Science, Cornell University, Ithaca NY, USA
– sequence: 2
  givenname: András
  surname: Sebő
  fullname: Sebő, András
  email: andras.sebo@grenoble-inp.fr
  organization: Optimisation Combinatoire (G-SCOP), CNRS, Univ. Grenoble Alpes, Grenoble, France
– sequence: 3
  givenname: Vera
  surname: Traub
  fullname: Traub, Vera
  email: traub@or.uni-bonn.de
  organization: Research Institute for Discrete Mathematics, University of Bonn, Bonn, Germany
– sequence: 4
  givenname: Anke
  surname: van Zuylen
  fullname: van Zuylen, Anke
  email: anke@wm.edu
  organization: Department of Mathematics, College of William & Mary, Williamsburg VA, USA
BackLink https://hal.science/hal-01592515$$DView record in HAL
BookMark eNp9kLFOwzAQhi1UJNrCA7BlZUi4Sxo7EVNVAUWKxAKz5dpn6iqNKzuq1I3X4PV4ElIVFoZOJ53-707_N2GjznfE2C1ChoD8fpP50GY5oMgQM4D8go2xEnkqZrwasfGQESkvhLhikxg3ACAqrMasatSBQkxUZ5Kt6oN3JibWh6RfU9IHtafWdR9JVC3Freq-P79islP9Ol6zS6vaSDe_c8renx7fFsu0eX1-WcybVBe86NPSGGPRFAa4URagXgFZWgmYYV1jrmlWKNA255zb0tqS6lrlSJWAkkzBdTFld6e7a9XKXXBbFQ7SKyeX80Yed4BlnZdY7nHIilNWBx9jICu161XvfDc0ca1EkEdZciMHWfIoSyLKQdZA4j_y79U55uHE0FB_7yjIqB11mowLpHtpvDtD_wDiwoT3
CitedBy_id crossref_primary_10_1016_j_orl_2020_09_009
crossref_primary_10_1016_j_orl_2019_02_005
crossref_primary_10_1145_3326123
Cites_doi 10.1007/978-3-319-33461-5_11
10.1016/j.orl.2013.08.006
10.1016/0095-8956(84)90023-6
10.1109/FOCS.2006.48
10.1109/FOCS.2016.21
10.1145/2818310
ContentType Journal Article
Copyright 2017 Elsevier B.V.
Distributed under a Creative Commons Attribution 4.0 International License
Copyright_xml – notice: 2017 Elsevier B.V.
– notice: Distributed under a Creative Commons Attribution 4.0 International License
DBID AAYXX
CITATION
1XC
DOI 10.1016/j.orl.2017.11.002
DatabaseName CrossRef
Hyper Article en Ligne (HAL)
DatabaseTitle CrossRef
DatabaseTitleList

DeliveryMethod fulltext_linktorsrc
Discipline Engineering
Sciences (General)
Mathematics
Computer Science
EISSN 1872-7468
EndPage 63
ExternalDocumentID oai:HAL:hal-01592515v1
10_1016_j_orl_2017_11_002
S0167637717302699
GroupedDBID --K
--M
-~X
.DC
.~1
0R~
123
1B1
1OL
1RT
1~.
1~5
29N
4.4
457
4G.
4R4
5VS
7-5
71M
8P~
9JN
9JO
AAAKF
AAAKG
AACTN
AAEDT
AAEDW
AAIAV
AAIKJ
AAKOC
AALRI
AAOAW
AAQFI
AAQXK
AARIN
AAXUO
ABAOU
ABFNM
ABJNI
ABMAC
ABUCO
ABXDB
ABYKQ
ACAZW
ACDAQ
ACGFS
ACNCT
ACRLP
ADBBV
ADEZE
ADGUI
ADIYS
ADMUD
AEBSH
AEKER
AENEX
AFFNX
AFKWA
AFTJW
AGHFR
AGUBO
AGYEJ
AHHHB
AIEXJ
AIGVJ
AIKHN
AITUG
AJBFU
AJOXV
ALMA_UNASSIGNED_HOLDINGS
AMFUW
AMRAJ
APLSM
ARUGR
ASPBG
AVWKF
AXJTR
AZFZN
BKOJK
BLXMC
CS3
DU5
EBS
EFJIC
EFLBG
EJD
EO8
EO9
EP2
EP3
FDB
FEDTE
FGOYB
FIRID
FNPLU
FYGXN
G-Q
GBLVA
HAMUX
HVGLF
HZ~
IHE
J1W
KOM
LY1
M41
MHUIS
MO0
MS~
N9A
O-L
O9-
OAUVE
OZT
P-8
P-9
P2P
PC.
Q38
R2-
RIG
ROL
RPZ
SDF
SDG
SDS
SES
SEW
SPC
SPCBC
SSB
SSD
SSW
SSZ
T5K
TN5
WH7
WUQ
XPP
XSW
~G-
9DU
AATTM
AAXKI
AAYWO
AAYXX
ABWVN
ACLOT
ACRPL
ACVFH
ADCNI
ADNMO
AEIPS
AEUPX
AFJKZ
AFPUW
AGQPQ
AIGII
AIIUN
AKBMS
AKRWK
AKYEP
ANKPU
APXCP
CITATION
EFKBS
~HD
1XC
ID FETCH-LOGICAL-c363t-5dddf1d3d06daf009b0efeb70419912ce43a0cf2666f5ff5e99a21e8705ed36c3
ISICitedReferencesCount 4
ISICitedReferencesURI http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000425203900011&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
ISSN 0167-6377
IngestDate Tue Oct 14 20:55:26 EDT 2025
Sat Nov 29 02:30:18 EST 2025
Tue Nov 18 22:15:41 EST 2025
Fri Feb 23 02:30:13 EST 2024
IsPeerReviewed true
IsScholarly true
Issue 1
Keywords Christofides’ heuristic
Polyhedra
Matroid partition
Approximation algorithm
Path traveling salesman problem (TSP)
Spanning tree
Language English
License Distributed under a Creative Commons Attribution 4.0 International License: http://creativecommons.org/licenses/by/4.0
LinkModel OpenURL
MergedId FETCHMERGED-LOGICAL-c363t-5dddf1d3d06daf009b0efeb70419912ce43a0cf2666f5ff5e99a21e8705ed36c3
PageCount 4
ParticipantIDs hal_primary_oai_HAL_hal_01592515v1
crossref_citationtrail_10_1016_j_orl_2017_11_002
crossref_primary_10_1016_j_orl_2017_11_002
elsevier_sciencedirect_doi_10_1016_j_orl_2017_11_002
PublicationCentury 2000
PublicationDate January 2018
2018-01-00
2018
PublicationDateYYYYMMDD 2018-01-01
PublicationDate_xml – month: 01
  year: 2018
  text: January 2018
PublicationDecade 2010
PublicationTitle Operations research letters
PublicationYear 2018
Publisher Elsevier B.V
Elsevier
Publisher_xml – name: Elsevier B.V
– name: Elsevier
References Gao (b4) 2013; 41
Edmonds (b3) 1968; vol. 11
A. Sebő, A. van Zuylen, The salesman’s improved paths: a 3/2 + 1/34 approximation, in: Proceedings of the IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS, 2016, pp. 118–127.
C. Gottschalk, J. Vygen, Better s-t-tours by Gao trees, in: Proceedings of the 18th International Conference on Integer Programming and Combinatorial Optimization, IPCO, 2016, pp. 126–137.
Cunningham (b2) 1984; 36
Schrijver (b7) 2003
Traub, Vygen (b9) 2018
M. Goemans, Minimum bounded degree spanning trees, in: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, 2006,pp. 273–282.
An, Kleinberg, Shmoys (b1) 2015; 62
Traub (10.1016/j.orl.2017.11.002_b9) 2018
Schrijver (10.1016/j.orl.2017.11.002_b7) 2003
Gao (10.1016/j.orl.2017.11.002_b4) 2013; 41
10.1016/j.orl.2017.11.002_b5
10.1016/j.orl.2017.11.002_b6
Edmonds (10.1016/j.orl.2017.11.002_b3) 1968; vol. 11
10.1016/j.orl.2017.11.002_b8
Cunningham (10.1016/j.orl.2017.11.002_b2) 1984; 36
An (10.1016/j.orl.2017.11.002_b1) 2015; 62
References_xml – volume: 62
  start-page: 34
  year: 2015
  ident: b1
  article-title: Improving Christofides’ algorithm for the s-t path TSP
  publication-title: J. ACM
– reference: A. Sebő, A. van Zuylen, The salesman’s improved paths: a 3/2 + 1/34 approximation, in: Proceedings of the IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS, 2016, pp. 118–127.
– volume: 36
  start-page: 161
  year: 1984
  end-page: 188
  ident: b2
  article-title: Testing membership in matroid polyhedra
  publication-title: J. Combin. Theory Ser. B
– year: 2003
  ident: b7
  publication-title: Combinatorial Optimization - Polyhedra and Efficiency
– year: 2018
  ident: b9
  article-title: Approaching
  publication-title: Proceedings of the ACM-SIAM Symposium on Discrete Algorithms, SODA
– volume: 41
  start-page: 615
  year: 2013
  end-page: 617
  ident: b4
  article-title: An LP-based 3/2-approximation algorithm for the s-t path graph Traveling Salesman Problem
  publication-title: Oper. Res. Lett.
– reference: M. Goemans, Minimum bounded degree spanning trees, in: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, 2006,pp. 273–282.
– volume: vol. 11
  start-page: 335
  year: 1968
  end-page: 345
  ident: b3
  publication-title: Matroid Partition
– reference: C. Gottschalk, J. Vygen, Better s-t-tours by Gao trees, in: Proceedings of the 18th International Conference on Integer Programming and Combinatorial Optimization, IPCO, 2016, pp. 126–137.
– ident: 10.1016/j.orl.2017.11.002_b6
  doi: 10.1007/978-3-319-33461-5_11
– volume: 41
  start-page: 615
  issue: 6
  year: 2013
  ident: 10.1016/j.orl.2017.11.002_b4
  article-title: An LP-based 3/2-approximation algorithm for the s-t path graph Traveling Salesman Problem
  publication-title: Oper. Res. Lett.
  doi: 10.1016/j.orl.2013.08.006
– volume: 36
  start-page: 161
  year: 1984
  ident: 10.1016/j.orl.2017.11.002_b2
  article-title: Testing membership in matroid polyhedra
  publication-title: J. Combin. Theory Ser. B
  doi: 10.1016/0095-8956(84)90023-6
– year: 2018
  ident: 10.1016/j.orl.2017.11.002_b9
  article-title: Approaching 32 for the s-t-path TSP
– ident: 10.1016/j.orl.2017.11.002_b5
  doi: 10.1109/FOCS.2006.48
– volume: vol. 11
  start-page: 335
  year: 1968
  ident: 10.1016/j.orl.2017.11.002_b3
– ident: 10.1016/j.orl.2017.11.002_b8
  doi: 10.1109/FOCS.2016.21
– year: 2003
  ident: 10.1016/j.orl.2017.11.002_b7
– volume: 62
  start-page: 34
  issue: 5
  year: 2015
  ident: 10.1016/j.orl.2017.11.002_b1
  article-title: Improving Christofides’ algorithm for the s-t path TSP
  publication-title: J. ACM
  doi: 10.1145/2818310
SSID ssj0007818
Score 2.198994
Snippet Gottschalk and Vygen proved that every solution of the subtour elimination linear program for traveling salesman paths is a convex combination of more and more...
Gottschalk and Vygen proved that every solution of the well-known subtour elimination linear program for traveling salesman paths is a convex combination of a...
SourceID hal
crossref
elsevier
SourceType Open Access Repository
Enrichment Source
Index Database
Publisher
StartPage 60
SubjectTerms Approximation algorithm
Christofides’ heuristic
Combinatorics
Computer Science
Discrete Mathematics
Mathematics
Matroid partition
Path traveling salesman problem (TSP)
Polyhedra
Spanning tree
Title Layers and matroids for the traveling salesman’s paths
URI https://dx.doi.org/10.1016/j.orl.2017.11.002
https://hal.science/hal-01592515
Volume 46
WOSCitedRecordID wos000425203900011&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
hasFullText 1
inHoldings 1
isFullTextHit
isPrint
journalDatabaseRights – providerCode: PRVESC
  databaseName: Elsevier SD Freedom Collection Journals 2021
  customDbUrl:
  eissn: 1872-7468
  dateEnd: 99991231
  omitProxy: false
  ssIdentifier: ssj0007818
  issn: 0167-6377
  databaseCode: AIEXJ
  dateStart: 19950201
  isFulltext: true
  titleUrlDefault: https://www.sciencedirect.com
  providerName: Elsevier
link http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV3db9MwELdKxwM8ACsgxpciBOJLmZI4iZ3HCjZ1UMqklaniJXJiW91W2ippq_Fv8Bdzjj8WQEzjgZeoTWwr9f169_P5fIfQcxwFMilI4rOCwgIlLbjPWIp9kRUlj3FcyKby3PGQjEZ0MskOO50f9izMZkbmc3p-ni3_q6jhHghbHZ39B3G7QeEGfAahwxXEDtcrCX7IFIs2ARSranHCaxdLOFbFhnQSbjAMdeP9JLXK0z-t2zT181JUJkbOZAOavp01B38cBT8qpzDEGfu2dPzXPRLAUZMXWWgjJpvd-NA9h7dYN9tAx6JyVkGdo_q6_j4TJqfBmWg7JIz2bPxjxpi33ZWghlNsCrUIrWIpAU4f62I6VgfHbSWqCwwYc6zV3x-KXvscTncXldo_CsmuSsUaRBdWze7kD_pH-eH7_Xx4MPr469NWJOKgP4QrTJsP7CgDypdsYEW9FZEkC7poq3-wN_ng7DuhjdfY_Ta7V95EDf72Pn9jO9em1m_f8JjxHXTLLEC8vgbONuqIeQ_dtsU9PKPre-hmK1MlfPvk0vvWPbRtWtXeK5O2_PVdlGrgeQA8zwLPA-B50NFzwPMs8F7WXgO7e-jL_t743cA3VTn8Eqd45SeccxlyzIOUMwkUvQiEFAUJYhVFF5UixiwoJRC_VCZSJiLLWBQKsAuJ4Dgt8X3UnS_m4gHyQk55FsSM0pjHkhUZZZSUwOkli4ikeAcFdvby0qSsV5VTZrmNTTzNYcJzNeGwlM1hwnfQG9dlqfO1XNY4tiLJDeHURDIHmF3W7RmIzw2vErQDgHJ17wI-D6_S6BG6of5B2qX3GHVX1Vo8QdfLzeqkrp4a4P0EhpKnCA
linkProvider Elsevier
openUrl ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Layers+and+Matroids+for+the+Traveling+Salesman%27s+Paths&rft.jtitle=Operations+research+letters&rft.au=Schalekamp%2C+Frans&rft.au=Seb%C5%91%2C+Andr%C3%A1s&rft.au=Traub%2C+Vera&rft.au=van+Zuylen%2C+Anke&rft.date=2018&rft.pub=Elsevier&rft.issn=0167-6377&rft.eissn=1872-7468&rft.volume=46&rft.spage=60&rft.epage=63&rft_id=info:doi/10.1016%2Fj.orl.2017.11.002&rft.externalDBID=HAS_PDF_LINK&rft.externalDocID=oai%3AHAL%3Ahal-01592515v1
thumbnail_l http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0167-6377&client=summon
thumbnail_m http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0167-6377&client=summon
thumbnail_s http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0167-6377&client=summon