Polynomial-Time Approximation Algorithm for the Problem of Cardinality-Weighted Variance-Based 2-Clustering with a Given Center
A strongly NP-hard problem of partitioning a finite set of points of Euclidean space into two clusters is considered. The solution criterion is the minimum of the sum (over both clusters) of weighted sums of squared distances from the elements of each cluster to its geometric center. The weights of...
Saved in:
| Published in: | Computational mathematics and mathematical physics Vol. 58; no. 1; pp. 130 - 136 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Moscow
Pleiades Publishing
01.01.2018
Springer Nature B.V |
| Subjects: | |
| ISSN: | 0965-5425, 1555-6662 |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Abstract | A strongly NP-hard problem of partitioning a finite set of points of Euclidean space into two clusters is considered. The solution criterion is the minimum of the sum (over both clusters) of weighted sums of squared distances from the elements of each cluster to its geometric center. The weights of the sums are equal to the cardinalities of the desired clusters. The center of one cluster is given as input, while the center of the other is unknown and is determined as the point of space equal to the mean of the cluster elements. A version of the problem is analyzed in which the cardinalities of the clusters are given as input. A polynomial-time 2-approximation algorithm for solving the problem is constructed. |
|---|---|
| AbstractList | A strongly NP-hard problem of partitioning a finite set of points of Euclidean space into two clusters is considered. The solution criterion is the minimum of the sum (over both clusters) of weighted sums of squared distances from the elements of each cluster to its geometric center. The weights of the sums are equal to the cardinalities of the desired clusters. The center of one cluster is given as input, while the center of the other is unknown and is determined as the point of space equal to the mean of the cluster elements. A version of the problem is analyzed in which the cardinalities of the clusters are given as input. A polynomial-time 2-approximation algorithm for solving the problem is constructed. |
| Author | Motkova, A. V. Kel’manov, A. V. |
| Author_xml | – sequence: 1 givenname: A. V. surname: Kel’manov fullname: Kel’manov, A. V. email: kelm@math.nsc.ru organization: Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, Novosibirsk State University – sequence: 2 givenname: A. V. surname: Motkova fullname: Motkova, A. V. organization: Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, Novosibirsk State University |
| BookMark | eNp9kMFqGzEQhkVIoE7SB-hNkLNaaXclrY7OkqYBQwMxyXHRrmZtmV3JkeSmPvXVK8eFQkNzEqP5v2HmO0enzjtA6BOjnxkrqy8PVAnOq4KzmjJKZXWCZoxzToQQxSmaHdrk0P-AzmPcUMqEqssZ-nXvx73zk9UjWdoJ8Hy7Df6nnXSy3uH5uPLBpvWEBx9wWgO-D74bYcJ-wI0Oxjo92rQnT2BX6wQGP-pgteuBXOuYy4I04y4mCNat8EuehDW-tT_A4QZc_r5EZ4MeI3z8816g5debZfONLL7f3jXzBelLJhLhoOoOSiOMFgBKyE6JSkpQIE3BKO_5oMquq0SXI9VgKAiTD1RGmq4TvLxAV8ex-bjnHcTUbvwu5N1jW1CqaiaFrHNKHlN98DEGGNreplcRKWg7toy2B9ntG9mZZP-Q25Adhv27THFk4vZgB8Lfnf4P_QaYS5N8 |
| CitedBy_id | crossref_primary_10_1134_S0965542519050099 |
| Cites_doi | 10.1007/978-3-642-95322-4_5 10.1134/S096554251502013X 10.1134/S0081543816090066 10.1134/S0965542516020111 10.1134/S1990478915040067 10.1080/01621459.1971.10482319 10.1134/S1990478916030054 10.1134/S1990478913040066 10.1134/S096554251603009X 10.1134/S1990478911040107 10.1134/S1990478914030041 10.1134/S1064562415050233 10.1007/978-3-319-14142-8 10.1007/978-3-319-44914-2_15 10.1145/321958.321975 10.1007/s10994-009-5103-0 10.1007/978-0-387-21606-5 |
| ContentType | Journal Article |
| Copyright | Pleiades Publishing, Ltd. 2018 Computational Mathematics and Mathematical Physics is a copyright of Springer, (2018). All Rights Reserved. |
| Copyright_xml | – notice: Pleiades Publishing, Ltd. 2018 – notice: Computational Mathematics and Mathematical Physics is a copyright of Springer, (2018). All Rights Reserved. |
| DBID | AAYXX CITATION 3V. 7SC 7TB 7U5 7WY 7WZ 7XB 87Z 88I 8AL 8FD 8FE 8FG 8FK 8FL ABJCF ABUWG AFKRA ARAPS AZQEC BENPR BEZIV BGLVJ CCPQU DWQXO FR3 FRNLG F~G GNUQQ HCIFZ JQ2 K60 K6~ K7- KR7 L.- L6V L7M L~C L~D M0C M0N M2P M7S P5Z P62 PHGZM PHGZT PKEHL PQBIZ PQBZA PQEST PQGLB PQQKQ PQUKI PRINS PTHSS PYYUZ Q9U |
| DOI | 10.1134/S0965542518010074 |
| DatabaseName | CrossRef ProQuest Central (Corporate) Computer and Information Systems Abstracts Mechanical & Transportation Engineering Abstracts Solid State and Superconductivity Abstracts ABI/INFORM Collection ABI/INFORM Global (PDF only) ProQuest Central (purchase pre-March 2016) ABI/INFORM Global (Alumni Edition) Science Database (Alumni Edition) Computing Database (Alumni Edition) Technology Research Database ProQuest SciTech Collection ProQuest Technology Collection ProQuest Central (Alumni) (purchase pre-March 2016) ABI/INFORM Collection (Alumni Edition) ProQuest Materials Science & Engineering Collection ProQuest Central (Alumni) ProQuest Central UK/Ireland Advanced Technologies & Computer Science Collection ProQuest Central Essentials AUTh Library subscriptions: ProQuest Central Business Premium Collection Technology collection ProQuest One Community College ProQuest Central Engineering Research Database Business Premium Collection (Alumni) ABI/INFORM Global (Corporate) ProQuest Central Student SciTech Premium Collection ProQuest Computer Science Collection ProQuest Business Collection (Alumni Edition) ProQuest Business Collection Computer Science Database Civil Engineering Abstracts ABI/INFORM Professional Advanced ProQuest Engineering Collection Advanced Technologies Database with Aerospace Computer and Information Systems Abstracts Academic Computer and Information Systems Abstracts Professional ABI/INFORM Global (OCUL) Computing Database Science Database ProQuest Engineering Database (NC LIVE) ProQuest advanced technologies & aerospace journals ProQuest Advanced Technologies & Aerospace Collection ProQuest Databases ProQuest One Academic (New) ProQuest One Academic Middle East (New) ProQuest One Business ProQuest One Business (Alumni) ProQuest One Academic Eastern Edition (DO NOT USE) ProQuest One Applied & Life Sciences ProQuest One Academic (retired) ProQuest One Academic UKI Edition ProQuest Central China Engineering Collection ABI/INFORM Collection China ProQuest Central Basic |
| DatabaseTitle | CrossRef ProQuest Business Collection (Alumni Edition) Computer Science Database ProQuest Central Student ProQuest Advanced Technologies & Aerospace Collection ProQuest Central Essentials ProQuest Computer Science Collection Computer and Information Systems Abstracts SciTech Premium Collection ProQuest Central China ABI/INFORM Complete ProQuest One Applied & Life Sciences ProQuest Central (New) Engineering Collection Advanced Technologies & Aerospace Collection Business Premium Collection ABI/INFORM Global Engineering Database ProQuest Science Journals (Alumni Edition) ProQuest One Academic Eastern Edition ProQuest Technology Collection ProQuest Business Collection ProQuest One Academic UKI Edition Solid State and Superconductivity Abstracts Engineering Research Database ProQuest One Academic ProQuest One Academic (New) ABI/INFORM Global (Corporate) ProQuest One Business Technology Collection Technology Research Database Computer and Information Systems Abstracts – Academic ProQuest One Academic Middle East (New) Mechanical & Transportation Engineering Abstracts ProQuest Central (Alumni Edition) ProQuest One Community College ProQuest Central ABI/INFORM Professional Advanced ProQuest Engineering Collection ProQuest Central Korea Advanced Technologies Database with Aerospace ABI/INFORM Complete (Alumni Edition) Civil Engineering Abstracts ProQuest Computing ABI/INFORM Global (Alumni Edition) ProQuest Central Basic ProQuest Science Journals ProQuest Computing (Alumni Edition) ABI/INFORM China ProQuest SciTech Collection Computer and Information Systems Abstracts Professional Advanced Technologies & Aerospace Database Materials Science & Engineering Collection ProQuest One Business (Alumni) ProQuest Central (Alumni) Business Premium Collection (Alumni) |
| DatabaseTitleList | ProQuest Business Collection (Alumni Edition) |
| Database_xml | – sequence: 1 dbid: BENPR name: ProQuest Central url: https://www.proquest.com/central sourceTypes: Aggregation Database |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Applied Sciences Mathematics |
| EISSN | 1555-6662 |
| EndPage | 136 |
| ExternalDocumentID | 10_1134_S0965542518010074 |
| GroupedDBID | --K -5D -5G -BR -EM -Y2 -~C -~X .VR 06D 0R~ 0VY 1B1 1N0 29F 2J2 2JN 2JY 2KG 2KM 2LR 2VQ 2~H 30V 3V. 4.4 408 40D 40E 5GY 5VS 6J9 6NX 7WY 88I 8FE 8FG 8FH 8FL 8TC 95- 95. 95~ 96X AAAVM AABHQ AACDK AAHNG AAIAL AAJBT AAJKR AANZL AARHV AARTL AASML AATNV AATVU AAUYE AAWCG AAYIU AAYQN AAYTO AAYZH ABAKF ABBBX ABDZT ABECU ABEFU ABFTV ABHLI ABHQN ABJCF ABJNI ABJOX ABKCH ABKTR ABMNI ABMQK ABNWP ABQBU ABQSL ABSXP ABTEG ABTHY ABTKH ABTMW ABULA ABUWG ABWNU ABXPI ACAOD ACBXY ACDTI ACGFS ACGOD ACHSB ACHXU ACIWK ACKNC ACMDZ ACMLO ACOKC ACOMO ACPIV ACREN ACSNA ACZOJ ADHHG ADHIR ADINQ ADKNI ADKPE ADRFC ADTPH ADURQ ADYFF ADYOE ADZKW AEBTG AEFQL AEGAL AEGNC AEJHL AEJRE AEKMD AEMSY AENEX AEOHA AEPYU AETLH AEVLU AEXYK AFBBN AFFNX AFGCZ AFKRA AFLOW AFQWF AFWTZ AFYQB AFZKB AGAYW AGDGC AGJBK AGMZJ AGQMX AGRTI AGWIL AGWZB AGYKE AHAVH AHBYD AHKAY AHSBF AHYZX AIAKS AIGIU AIIXL AILAN AITGF AJBLW AJRNO ALMA_UNASSIGNED_HOLDINGS ALWAN AMKLP AMTXH AMXSW AMYLF AMYQR AOCGG ARAPS ARMRJ ASPBG AVWKF AXYYD AZFZN AZQEC B-. BA0 BAPOH BDATZ BENPR BEZIV BGLVJ BGNMA BPHCQ BSONS CAG CCPQU COF CS3 CSCUP DDRTE DNIVK DPUIP DU5 DWQXO EBLON EBS EIOEI EJD ESBYG FEDTE FERAY FFXSO FIGPU FINBP FNLPD FRNLG FRRFC FSGXE FWDCC GGCAI GGRSB GJIRD GNUQQ GNWQR GQ6 GQ7 GROUPED_ABI_INFORM_COMPLETE H13 HCIFZ HF~ HG6 HLICF HMJXF HRMNR HVGLF HZ~ H~9 I-F IHE IJ- IKXTQ IWAJR IXD I~X I~Z J-C JBSCW JZLTJ K60 K6V K6~ K7- KOV L6V LK5 LLZTM M0C M0N M2P M41 M4Y M7R M7S MA- MK~ N2Q NB0 NPVJJ NQ- NQJWS NU0 O9- O93 O9J P62 P9R PF0 PQBIZ PQBZA PQQKQ PROAC PT4 PTHSS Q2X QOS R89 R9I RIG RNS ROL RPZ RSV S16 S1Z S27 S3B SAP SDH SHX SISQX SJYHP SMT SNE SNPRN SNX SOHCF SOJ SPISZ SRMVM SSLCW STPWE SZN T13 TSG TUC TUS UG4 UOJIU UTJUX UZXMN VC2 VFIZW VOH W23 W48 WK8 XPP XU3 YLTOR ZMTXR ~A9 AAPKM AAYXX ABDBE ABFSG ABRTQ ACSTC ADHKG AEZWR AFDZB AFFHD AFHIU AFOHR AGQPQ AHPBZ AHWEU AIXLP AMVHM ATHPR CITATION PHGZM PHGZT PQGLB 7SC 7TB 7U5 7XB 8AL 8FD 8FK FR3 JQ2 KR7 L.- L7M L~C L~D PKEHL PQEST PQUKI PRINS Q9U |
| ID | FETCH-LOGICAL-c316t-5e98be3d6da6ee967b96477e9e7d2105c5f93bb46bd6d4fd0e6d6989d7dbb653 |
| IEDL.DBID | M2P |
| ISICitedReferencesCount | 5 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000426674100009&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 0965-5425 |
| IngestDate | Tue Oct 07 13:11:54 EDT 2025 Tue Nov 18 22:25:21 EST 2025 Sat Nov 29 05:16:06 EST 2025 Fri Feb 21 02:37:59 EST 2025 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 1 |
| Keywords | NP-hardness Euclidean space weighted 2-clustering polynomial-time 2-approximation algorithm |
| Language | English |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c316t-5e98be3d6da6ee967b96477e9e7d2105c5f93bb46bd6d4fd0e6d6989d7dbb653 |
| Notes | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| PQID | 2009817678 |
| PQPubID | 60276 |
| PageCount | 7 |
| ParticipantIDs | proquest_journals_2009817678 crossref_citationtrail_10_1134_S0965542518010074 crossref_primary_10_1134_S0965542518010074 springer_journals_10_1134_S0965542518010074 |
| PublicationCentury | 2000 |
| PublicationDate | 20180100 2018-1-00 20180101 |
| PublicationDateYYYYMMDD | 2018-01-01 |
| PublicationDate_xml | – month: 1 year: 2018 text: 20180100 |
| PublicationDecade | 2010 |
| PublicationPlace | Moscow |
| PublicationPlace_xml | – name: Moscow |
| PublicationTitle | Computational mathematics and mathematical physics |
| PublicationTitleAbbrev | Comput. Math. and Math. Phys |
| PublicationYear | 2018 |
| Publisher | Pleiades Publishing Springer Nature B.V |
| Publisher_xml | – name: Pleiades Publishing – name: Springer Nature B.V |
| References | Wirth (CR25) 1976 Dolgushev, Kel’manov, Shenmaier (CR19) 2016; 1 Brucker (CR10) 1978; 157 Dolgushev, Kel’manov (CR18) 2011; 5 Aggarwal (CR5) 2015 Kel’manov, Motkova (CR4) 2016; 9869 Garey, Johnson (CR8) 1979 Kel’manov, Khandeev (CR22) 2015; 9 Fisher (CR15) 1956 Kel’manov, Khandeev (CR23) 2016; 56 de la Vega, Karpinski, Kenyon, Rabani (CR14) 2002 Sahni, Gonzalez (CR9) 1976; 23 Rao (CR16) 1971; 66 Kel’manov, Motkova (CR3) 2016; 10 de la Vega, Kenyon (CR13) 2001; 63 Hasegawa, Imai, Inaba, Katoh, Nakano (CR12) 1993; 1 Inaba, Katoh, Imai (CR11) 1994 Bishop (CR6) 2006 Kel’manov, Khandeev (CR20) 2013; 7 Kel’manov, Pyatkin (CR2) 2015; 56 Kel’manov, Pyatkin (CR1) 2015; 92 Kel’manov, Khandeev (CR21) 2015; 55 Aloise, Deshpande, Hansen, Popat (CR17) 2009; 75 Hastie, Tibshirani, Friedman (CR7) 2001 Kel’manov, Romanchenko (CR24) 2014; 8 A. V. Dolgushev (584_CR19) 2016; 1 R. A. Fisher (584_CR15) 1956 N. Wirth (584_CR25) 1976 A. V. Kel’manov (584_CR20) 2013; 7 A. V. Kel’manov (584_CR2) 2015; 56 A. V. Kel’manov (584_CR4) 2016; 9869 M. R. Garey (584_CR8) 1979 M. C. Bishop (584_CR6) 2006 M. Rao (584_CR16) 1971; 66 A. V. Kel’manov (584_CR21) 2015; 55 S. Hasegawa (584_CR12) 1993; 1 A. V. Kel’manov (584_CR1) 2015; 92 F. Vega de la (584_CR14) 2002 A. V. Kel’manov (584_CR23) 2016; 56 T. Hastie (584_CR7) 2001 C. C. Aggarwal (584_CR5) 2015 P. Brucker (584_CR10) 1978; 157 M. Inaba (584_CR11) 1994 D. Aloise (584_CR17) 2009; 75 F. Vega de la (584_CR13) 2001; 63 A. V. Kel’manov (584_CR24) 2014; 8 S. Sahni (584_CR9) 1976; 23 A. V. Dolgushev (584_CR18) 2011; 5 A. V. Kel’manov (584_CR3) 2016; 10 A. V. Kel’manov (584_CR22) 2015; 9 |
| References_xml | – year: 1979 ident: CR8 publication-title: Computers and Intractability: A Guide to the Theory of NP-Completeness – volume: 157 start-page: 45 year: 1978 end-page: 54 ident: CR10 article-title: On the complexity of clustering problems publication-title: Lect. Notes Econ. Math. Syst. doi: 10.1007/978-3-642-95322-4_5 – year: 2002 ident: CR14 article-title: Polynomial time approximation schemes for metric min-sum clustering publication-title: Electronic Colloquium on Computational Complexity (ECCC) – year: 1976 ident: CR25 publication-title: Algorithms + Data Structures = Programs – volume: 55 start-page: 330 issue: 2 year: 2015 end-page: 339 ident: CR21 article-title: A randomized algorithm for two-cluster partition of a set of vectors publication-title: Comput. Math. Math. Phys. doi: 10.1134/S096554251502013X – year: 1956 ident: CR15 publication-title: Statistical Methods and Scientific Inference – volume: 1 start-page: 47 year: 2016 end-page: 56 ident: CR19 article-title: A polynomial-time approximation scheme for a problem of partitioning a finite set into two clusters publication-title: Proc. Steklov Inst. Math. doi: 10.1134/S0081543816090066 – volume: 56 start-page: 334 issue: 2 year: 2016 end-page: 341 ident: CR23 article-title: Fully polynomial-time approximation scheme for a special case of a quadratic Euclidean 2-clustering problem publication-title: Comput. Math. Math. Phys. doi: 10.1134/S0965542516020111 – volume: 9 start-page: 497 issue: 4 year: 2015 end-page: 502 ident: CR22 article-title: An exact pseudopolynomial algorithm for a problem of the two-cluster partitioning of a set of vectors publication-title: J. Appl. Ind. Math. doi: 10.1134/S1990478915040067 – volume: 66 start-page: 622 year: 1971 end-page: 626 ident: CR16 article-title: Cluster analysis and mathematical programming publication-title: J. Am. Stat. Assoc. doi: 10.1080/01621459.1971.10482319 – start-page: 332 year: 1994 end-page: 339 ident: CR11 article-title: Applications of weighted Voronoi diagrams and randomization to variancebased k-clustering (extended abstract) publication-title: Proceedings of the 10th ACM Symposium on Computational Geometry – volume: 10 start-page: 349 issue: 3 year: 2016 end-page: 355 ident: CR3 article-title: Exact pseudopolynomial algorithm for a balanced 2-clustering problem publication-title: J. Appl. Ind. Math. doi: 10.1134/S1990478916030054 – volume: 7 start-page: 515 issue: 4 year: 2013 end-page: 521 ident: CR20 article-title: A 2-approximation polynomial algorithm for a clustering problem publication-title: J. Appl. Ind. Math. doi: 10.1134/S1990478913040066 – year: 2006 ident: CR6 publication-title: Pattern Recognition and Machine Learning – volume: 1 start-page: 75 year: 1993 end-page: 89 ident: CR12 article-title: Efficient algorithms for variance-based k-clustering publication-title: Proceedings of the 1st Pacific Conference on Computer Graphics and Applications – volume: 56 start-page: 491 issue: 3 year: 2015 end-page: 497 ident: CR2 article-title: On the complexity of some quadratic Euclidean 2-clustering problems publication-title: Comput. Math. Math. Phys. doi: 10.1134/S096554251603009X – volume: 5 start-page: 551 issue: 4 year: 2011 end-page: 558 ident: CR18 article-title: An approximation algorithm for solving a problem of cluster analysis publication-title: J. Appl. Ind. Math. doi: 10.1134/S1990478911040107 – volume: 8 start-page: 329 issue: 3 year: 2014 end-page: 336 ident: CR24 article-title: An FPTAS for a vector subset search problem publication-title: J. Appl. Ind. Math. doi: 10.1134/S1990478914030041 – volume: 92 start-page: 634 issue: 2 year: 2015 end-page: 637 ident: CR1 article-title: NP-hardness of some quadratic Euclidean 2-clustering problems publication-title: Dokl. Math. doi: 10.1134/S1064562415050233 – year: 2015 ident: CR5 publication-title: Data Mining doi: 10.1007/978-3-319-14142-8 – volume: 9869 start-page: 182 year: 2016 end-page: 192 ident: CR4 article-title: A fully polynomial-time approximation scheme for a special case of a balanced 2-clustering problem publication-title: Lect. Notes Comput. Sci. doi: 10.1007/978-3-319-44914-2_15 – volume: 63 start-page: 531 year: 2001 end-page: 541 ident: CR13 article-title: A randomized approximation scheme for metric max-cut publication-title: J. Comput. Sci. – volume: 23 start-page: 555 year: 1976 end-page: 566 ident: CR9 article-title: P-complete approximation problems publication-title: J. ACM doi: 10.1145/321958.321975 – volume: 75 start-page: 245 issue: 2 year: 2009 end-page: 248 ident: CR17 article-title: NP-hardness of Euclidean sum-of-squares clustering publication-title: Machine Learning doi: 10.1007/s10994-009-5103-0 – year: 2001 ident: CR7 publication-title: The Elements of Statistical Learning: Data Mining, Inference, and Prediction doi: 10.1007/978-0-387-21606-5 – volume: 9 start-page: 497 issue: 4 year: 2015 ident: 584_CR22 publication-title: J. Appl. Ind. Math. doi: 10.1134/S1990478915040067 – volume: 1 start-page: 75 year: 1993 ident: 584_CR12 publication-title: Proceedings of the 1st Pacific Conference on Computer Graphics and Applications – volume: 9869 start-page: 182 year: 2016 ident: 584_CR4 publication-title: Lect. Notes Comput. Sci. doi: 10.1007/978-3-319-44914-2_15 – volume-title: Data Mining year: 2015 ident: 584_CR5 doi: 10.1007/978-3-319-14142-8 – volume: 5 start-page: 551 issue: 4 year: 2011 ident: 584_CR18 publication-title: J. Appl. Ind. Math. doi: 10.1134/S1990478911040107 – volume-title: The Elements of Statistical Learning: Data Mining, Inference, and Prediction year: 2001 ident: 584_CR7 doi: 10.1007/978-0-387-21606-5 – volume: 1 start-page: 47 year: 2016 ident: 584_CR19 publication-title: Proc. Steklov Inst. Math. doi: 10.1134/S0081543816090066 – volume: 7 start-page: 515 issue: 4 year: 2013 ident: 584_CR20 publication-title: J. Appl. Ind. Math. doi: 10.1134/S1990478913040066 – volume: 8 start-page: 329 issue: 3 year: 2014 ident: 584_CR24 publication-title: J. Appl. Ind. Math. doi: 10.1134/S1990478914030041 – volume: 66 start-page: 622 year: 1971 ident: 584_CR16 publication-title: J. Am. Stat. Assoc. doi: 10.1080/01621459.1971.10482319 – volume: 56 start-page: 491 issue: 3 year: 2015 ident: 584_CR2 publication-title: Comput. Math. Math. Phys. doi: 10.1134/S096554251603009X – volume-title: Pattern Recognition and Machine Learning year: 2006 ident: 584_CR6 – volume: 63 start-page: 531 year: 2001 ident: 584_CR13 publication-title: J. Comput. Sci. – volume-title: Algorithms + Data Structures = Programs year: 1976 ident: 584_CR25 – volume: 23 start-page: 555 year: 1976 ident: 584_CR9 publication-title: J. ACM doi: 10.1145/321958.321975 – volume-title: Computers and Intractability: A Guide to the Theory of NP-Completeness year: 1979 ident: 584_CR8 – start-page: 332 volume-title: Proceedings of the 10th ACM Symposium on Computational Geometry year: 1994 ident: 584_CR11 – volume: 55 start-page: 330 issue: 2 year: 2015 ident: 584_CR21 publication-title: Comput. Math. Math. Phys. doi: 10.1134/S096554251502013X – volume: 10 start-page: 349 issue: 3 year: 2016 ident: 584_CR3 publication-title: J. Appl. Ind. Math. doi: 10.1134/S1990478916030054 – volume: 75 start-page: 245 issue: 2 year: 2009 ident: 584_CR17 publication-title: Machine Learning doi: 10.1007/s10994-009-5103-0 – volume-title: Electronic Colloquium on Computational Complexity (ECCC) year: 2002 ident: 584_CR14 – volume: 92 start-page: 634 issue: 2 year: 2015 ident: 584_CR1 publication-title: Dokl. Math. doi: 10.1134/S1064562415050233 – volume: 157 start-page: 45 year: 1978 ident: 584_CR10 publication-title: Lect. Notes Econ. Math. Syst. doi: 10.1007/978-3-642-95322-4_5 – volume-title: Statistical Methods and Scientific Inference year: 1956 ident: 584_CR15 – volume: 56 start-page: 334 issue: 2 year: 2016 ident: 584_CR23 publication-title: Comput. Math. Math. Phys. doi: 10.1134/S0965542516020111 |
| SSID | ssj0016983 |
| Score | 2.1165373 |
| Snippet | A strongly NP-hard problem of partitioning a finite set of points of Euclidean space into two clusters is considered. The solution criterion is the minimum of... |
| SourceID | proquest crossref springer |
| SourceType | Aggregation Database Enrichment Source Index Database Publisher |
| StartPage | 130 |
| SubjectTerms | Approximation Clustering Computational Mathematics and Numerical Analysis Euclidean geometry Euclidean space Mathematical analysis Mathematics Mathematics and Statistics Polynomials Ratios Sums |
| SummonAdditionalLinks | – databaseName: Springer Nature - Connect here FIRST to enable access dbid: RSV link: http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1LT8MwDLZgcIADjwFivJQDJ1DEuqZpcxwTjwOgiU1jt6ltXJhUNrRuCE78deI-NvGU4FwnbRMnduz4-wAOfRQE8aG59tyQC1tI7odY5ZZEo81SCV-nrCVX7s2N1-2qZl7HnRS33YuUZLpTZ7wj4qRFOCWOUTHLbKpk-eZhwVg7j_gabludaepAqgx700hzEs9Tmd928dEYzTzMT0nR1Nacr_7rK9dgJXctWT3ThXWYw0EZVnM3k-WLOCnD8vUUqjXZgLfmMH6l2mQ_5lQPwuqEMv7Sz0oaWT2-H47644dHZrxbZtqxZkZBw4YRa5B6ZZ48v0tjrOZFHXP6JlXip8ZAalbjjXhCaAzmLxhFfZnPLmiLZRRYxtEmtM_P2o1LnrMy8NC25Jg7qLwAbS21LxGVdIO0mBUVutqcH53QiZQdBEIGRkREuopSE0uldnUQSMfegtJgOMBtYJatnUigGTaBAiNXVWs6CnybEkO-FckKVIvZ6YU5YjkRZ8S99ORii96X0a7A0bTJUwbX8ZvwXjHlvXzlJkTLqTzLNTa8AsfFFM8e_9jZzp-kd2HJeF5eFsvZg9J4NMF9WAyfx_1kdJAq9DsLOu3o priority: 102 providerName: Springer Nature |
| Title | Polynomial-Time Approximation Algorithm for the Problem of Cardinality-Weighted Variance-Based 2-Clustering with a Given Center |
| URI | https://link.springer.com/article/10.1134/S0965542518010074 https://www.proquest.com/docview/2009817678 |
| Volume | 58 |
| WOSCitedRecordID | wos000426674100009&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: PRVAVX databaseName: Springer Nature - Connect here FIRST to enable access customDbUrl: eissn: 1555-6662 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0016983 issn: 0965-5425 databaseCode: RSV dateStart: 20060101 isFulltext: true titleUrlDefault: https://link.springer.com/search?facet-content-type=%22Journal%22 providerName: Springer Nature |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV3db9MwED-xjQdeNj5FYVR-4AlkLakdO35CXbWBBKuiddoGL1ESX2BSaLe2Q_DEv85d4q4CxF54OSmJYyW6893Pd747gJcFai7x4aVPbSW10kYWFUYyNkjSbJwufNu15IMdj9Pzc5cFh9siHKtc6cRWUftZxT7yPfbip7El3frm8kpy1yiOroYWGhuwRcgm5iNdR4PsJopgXFeG05lEJiScIaoZK7034Zt8LyYVzXb0d7u0Bpt_xEdbs3O4878ffB-2A-AUw05CHsAdnD6EnQA-RVjai0fwM5s1PzhHuWgk54WIIVcb_37RpTaKYfOZJl9--SoI5QpCjSLrWtGIWS1GLGYdopdnra-Vpj6lXTiLlNwnQ-nFQI6aa67KQL8p2PsrCvGWVa1gBzPOH8PJ4cHJ6J0M3RlkpWKzlAm6tETljS8MojO2bJNa0aH1tI9MqqR2qiy1KWmIrn2ExnO3Sm99WZpEPYHN6WyKT0HEyie1RmKLRo21ddHA12WhOEBUxLXpQbRiTV6FyuXcQKPJ2x2M0vlf3OzBq5tXLruyHbcN3l1xMA8reJGv2deD1ysZWD_-52TPbp_sOdwjyJV2Tpxd2FzOr_EF3K2-LS8W8z5s2LOPfdjaPxhnx3T13kqiR9Go3wo2UzshmiWfiB5PTn8B2M78Fw |
| linkProvider | ProQuest |
| linkToHtml | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMw1V1LT9tAEB4hqNRegL7UtAH20F5arRLb67V9qKoQoCDSKFKjNjdr7R0XJDehedBy4hfxH5nxg4iicuPQ63q9ktffznNnPoC3BhW3-LDShkEqlae0NCm2paOR0KwjZWzBWtIL-v1wNIoGK3BV18LwtcpaJhaC2k5SjpG3OIofOgHJ1k9nvySzRnF2tabQKGFxjBe_yWWbfTzao__7znUP9ofdQ1mxCsjUc_Rc-hiFCXpWW6MRIx0kRTEmRhhY8n_81M8iL0mUTmiKymwbtWWWRRvYJNFMEkESf01xYzG-KegObpIWNKtkrte-9OksVElUx1OtrzzIYw5pBFbbt9Xg0rb9Kx1baLmDjf9sfzZhvTKnRafE_1NYwfEz2KhMa1EJrtlzuBxM8guuwDa55KoX0eFe6n9Oy8JN0cl_0LfMT34KsuEF2cRiUBLtiEkmunyISn9Ffi8iybT0NzPlcBDKXTIDrHBlN19wzwnaVcGxbWHEZ1YkgsPnOH0Bw4fYhZewOp6M8RUIx7N-ppBQoFBhFkRt12aJ8Tj9ZZxMN6BdIyFOq77sTA-Sx4V_5qn4Dnga8P7mlbOyKcl9k5s1YOJKPs3iJVoa8KGG3PLxPxd7ff9iO_D4cPilF_eO-sdv4AkZl2EZrmrC6ny6wC14lJ7PT2fT7eLkCIgfGInXTMZTAw |
| linkToPdf | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV3dT9swED_xMU3sgQIDrXz6gachi6ZxnPixdCsgoKoEYrxFSXyBSlmLmjDBE_86vsRpxTYmTTzn7CT22Xe-8_1-APsRCoL40FwHfsKFKySPEmxxR6LRZqlEpEvWknO_3w9ubtTA8pzm9W33OiVZ1TQQStOoOLzXqeUgEYeXhFniGXVzzAZLVnAeFgXdo6fj-uX1NI0gVYXDaaQ5idu05l-7eG2YZt7mbwnS0u70Gu_-4hVYti4n61Q6sgpzOFqDhnU_mV3c-Rp8uphCuOaf4Xkwzp6oZjnKONWJsA6hjz8Oq1JH1slux5NhcfeTGa-XmXZsUFHTsHHKuqR2lYfPf5SxV_Oia3MqJxXjR8Zwatbm3eyBUBrMHzGKBrOIHdPWyyjgjJN1uOp9v-qecMvWwBPXkQX3UAUxulrqSCIq6cdlkSsq9LU5V3qJlyo3joWMjYhIdQulJvZK7es4lp67AQuj8Qi_AHNc7aUCzbAJFJj6qtXWaRy5lDCKnFQ2oVXPVJhYJHMi1MjC8kTjivCP0W7C12mT-wrG41_C2_X0h3ZF50TXqQLHN7a9CQf1dM8ev9nZ5n9J78HHwbdeeH7aP9uCJeOcBVW4ZxsWiskD7sCH5FcxzCe7pZ6_AH4Q-bA |
| 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=Polynomial-Time+Approximation+Algorithm+for+the+Problem+of+Cardinality-Weighted+Variance-Based+2-Clustering+with+a+Given+Center&rft.jtitle=Computational+mathematics+and+mathematical+physics&rft.au=Kel%E2%80%99manov%2C+A.+V.&rft.au=Motkova%2C+A.+V.&rft.date=2018-01-01&rft.issn=0965-5425&rft.eissn=1555-6662&rft.volume=58&rft.issue=1&rft.spage=130&rft.epage=136&rft_id=info:doi/10.1134%2FS0965542518010074&rft.externalDBID=n%2Fa&rft.externalDocID=10_1134_S0965542518010074 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0965-5425&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0965-5425&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0965-5425&client=summon |