Maximizing Nonmonotone Submodular Functions under Matroid or Knapsack Constraints
Submodular function maximization is a central problem in combinatorial optimization, generalizing many important problems including Max Cut in directed/undirected graphs and in hypergraphs, certain constraint satisfaction problems, maximum entropy sampling, and maximum facility location problems. Un...
Saved in:
| Published in: | SIAM journal on discrete mathematics Vol. 23; no. 4; pp. 2053 - 2078 |
|---|---|
| Main Authors: | , , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Philadelphia, PA
Society for Industrial and Applied Mathematics
01.01.2010
|
| Subjects: | |
| ISSN: | 0895-4801, 1095-7146 |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Abstract | Submodular function maximization is a central problem in combinatorial optimization, generalizing many important problems including Max Cut in directed/undirected graphs and in hypergraphs, certain constraint satisfaction problems, maximum entropy sampling, and maximum facility location problems. Unlike submodular minimization, submodular maximization is NP-hard. In this paper, we give the first constant-factor approximation algorithm for maximizing any nonnegative submodular function subject to multiple matroid or knapsack constraints. We emphasize that our results are for nonmonotone submodular functions. In particular, for any constant $k$, we present a $(\frac{1}{k+2+\frac{1}{k}+\epsilon})$-approximation for the submodular maximization problem under $k$ matroid constraints, and a $(\frac{1}{5}-\epsilon)$-approximation algorithm for this problem subject to $k$ knapsack constraints ($\epsilon>0$ is any constant). We improve the approximation guarantee of our algorithm to $\frac{1}{k+1+\frac{1}{k-1}+\epsilon}$ for $k\geq2$ partition matroid constraints. This idea also gives a $(\frac{1}{k+\epsilon})$-approximation for maximizing a monotone submodular function subject to $k\geq2$ partition matroids, which is an improvement over the previously best known guarantee of $\frac{1}{k+1}$. |
|---|---|
| AbstractList | Submodular function maximization is a central problem in combinatorial optimization, generalizing many important problems including Max Cut in directed/undirected graphs and in hypergraphs, certain constraint satisfaction problems, maximum entropy sampling, and maximum facility location problems. Unlike submodular minimization, submodular maximization is NP-hard. In this paper, we give the first constant-factor approximation algorithm for maximizing any nonnegative submodular function subject to multiple matroid or knapsack constraints. We emphasize that our results are for nonmonotone submodular functions. In particular, for any constant $k$, we present a $(\frac{1}{k+2+\frac{1}{k}+\epsilon})$-approximation for the submodular maximization problem under $k$ matroid constraints, and a $(\frac{1}{5}-\epsilon)$-approximation algorithm for this problem subject to $k$ knapsack constraints ($\epsilon>0$ is any constant). We improve the approximation guarantee of our algorithm to $\frac{1}{k+1+\frac{1}{k-1}+\epsilon}$ for $k\geq2$ partition matroid constraints. This idea also gives a $(\frac{1}{k+\epsilon})$-approximation for maximizing a monotone submodular function subject to $k\geq2$ partition matroids, which is an improvement over the previously best known guarantee of $\frac{1}{k+1}$. |
| Author | Nagarajan, Viswanath Sviridenko, Maxim Lee, Jon Mirrokni, Vahab S. |
| Author_xml | – sequence: 1 givenname: Jon surname: Lee fullname: Lee, Jon – sequence: 2 givenname: Vahab S. surname: Mirrokni fullname: Mirrokni, Vahab S. – sequence: 3 givenname: Viswanath surname: Nagarajan fullname: Nagarajan, Viswanath – sequence: 4 givenname: Maxim surname: Sviridenko fullname: Sviridenko, Maxim |
| BackLink | http://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=23421152$$DView record in Pascal Francis |
| BookMark | eNptkEtLAzEQgINUsK0e_AdB8OBhbR6bTfYoxarYKqKelzSbSOo2qUkW1F9vSqUH8TTDzDcPvhEYOO80AKcYXWJM-QTViDOECDoAQ4xqVnBcVgMwRCLnpUD4CIxiXCGEyxKzIXhayE-7tt_WvcEH79be-ZQ3wud-ufZt38kAZ71TyXoXYe9aHeBCpuBtC32A905uolTvcJrbKUjrUjwGh0Z2UZ_8xjF4nV2_TG-L-ePN3fRqXijKSCoMVYxrTgnXuNZVhZASbKna_JnALdOkJaYuaWnM0lScMGNykTMhBBaKS03H4Gy3dxP8R69jala-Dy6fbGrCKsowExk6_4VkVLIzQTplY7MJdi3DV0NoSTBmJHMXO04FH2PQZo9g1GzFNnuxmZ38YZVNcmtoa6D7Z-IHl797NQ |
| CODEN | SJDMEC |
| CitedBy_id | crossref_primary_10_1002_cpe_6575 crossref_primary_10_1287_moor_2018_0955 crossref_primary_10_1007_s10107_017_1154_3 crossref_primary_10_1109_TMC_2021_3127212 crossref_primary_10_1007_s11280_022_01047_w crossref_primary_10_1287_mnsc_2015_2254 crossref_primary_10_1007_s11704_024_40266_4 crossref_primary_10_1016_j_tcs_2022_09_029 crossref_primary_10_1287_moor_2022_0320 crossref_primary_10_1145_2746226 crossref_primary_10_1145_3184990 crossref_primary_10_1137_130929205 crossref_primary_10_1007_s11590_019_01430_z crossref_primary_10_1016_j_disopt_2015_07_003 crossref_primary_10_1007_s10107_020_01524_y crossref_primary_10_1007_s10898_021_01063_6 crossref_primary_10_1287_moor_2021_1172 crossref_primary_10_1109_TPWRS_2017_2691320 crossref_primary_10_1007_s10107_020_01570_6 crossref_primary_10_1016_j_tcs_2024_114469 crossref_primary_10_1016_j_ipl_2013_03_018 crossref_primary_10_1145_3638052 crossref_primary_10_1109_TITS_2024_3435969 crossref_primary_10_1007_s10107_020_01607_w crossref_primary_10_1017_S0960129521000372 crossref_primary_10_1145_3570615 crossref_primary_10_1016_j_ipl_2020_106010 crossref_primary_10_1109_TWC_2016_2613517 crossref_primary_10_1137_16M1105220 crossref_primary_10_1016_j_tcs_2016_11_009 crossref_primary_10_1145_3070685 crossref_primary_10_1016_j_orl_2018_11_002 crossref_primary_10_1145_3381525 crossref_primary_10_1002_net_21619 crossref_primary_10_1007_s10878_020_00662_5 crossref_primary_10_1016_j_tcs_2023_114017 crossref_primary_10_1016_j_ins_2024_120259 crossref_primary_10_1007_s10107_020_01614_x crossref_primary_10_1016_j_tcs_2023_114293 crossref_primary_10_1137_120888909 crossref_primary_10_1007_s10957_022_02145_5 crossref_primary_10_1007_s10878_023_01001_0 crossref_primary_10_1007_s10878_023_01021_w crossref_primary_10_1007_s10878_024_01232_9 crossref_primary_10_1109_TNNLS_2018_2885591 crossref_primary_10_1016_j_disopt_2017_01_003 crossref_primary_10_1109_TNSE_2020_2984658 crossref_primary_10_1109_TKDE_2017_2787757 crossref_primary_10_1287_moor_2014_0668 crossref_primary_10_1145_3606376_3593573 crossref_primary_10_1287_mnsc_2023_01726 crossref_primary_10_1016_j_jcss_2021_11_005 crossref_primary_10_1137_19M125515X crossref_primary_10_1137_110839655 crossref_primary_10_1109_TMC_2020_3043000 crossref_primary_10_1287_moor_2018_0965 crossref_primary_10_1007_s10878_021_00774_6 crossref_primary_10_1287_opre_2021_2188 crossref_primary_10_1109_TNET_2024_3423316 crossref_primary_10_1016_j_orl_2025_107295 crossref_primary_10_1109_TWC_2018_2858261 crossref_primary_10_1007_s10878_020_00672_3 crossref_primary_10_1287_moor_1100_0463 crossref_primary_10_26599_TST_2023_9010107 crossref_primary_10_1109_TITS_2023_3326954 |
| Cites_doi | 10.1137/S089548010036813X 10.1145/502090.502096 10.1145/227683.227684 10.1080/02664768700000020 10.1287/opre.46.5.655 10.1007/BFb0121195 10.1016/S0166-218X(99)00103-1 10.1007/s00037-006-0205-6 10.1287/mnsc.45.11.1539 10.1016/0166-218X(83)90040-9 10.1016/S0167-6377(03)00062-2 10.1016/0377-2217(95)00205-7 10.1145/502090.502098 10.1007/BF01588971 10.1007/s101070050055 10.1023/B:JOCO.0000038913.96607.c2 10.1137/0910022 10.1006/jctb.2000.1989 10.1007/s10107-006-0024-1 10.1287/mnsc.23.8.789 10.1145/285055.285059 10.1287/opre.43.4.684 10.1016/S0167-5060(08)70732-5 |
| ContentType | Journal Article |
| Copyright | 2015 INIST-CNRS [Copyright] © 2010 Society for Industrial and Applied Mathematics |
| Copyright_xml | – notice: 2015 INIST-CNRS – notice: [Copyright] © 2010 Society for Industrial and Applied Mathematics |
| DBID | AAYXX CITATION IQODW 3V. 7WY 7WZ 7X2 7XB 87Z 88A 88F 88I 88K 8AL 8FE 8FG 8FH 8FK 8FL 8G5 ABJCF ABUWG AFKRA ARAPS ATCPS AZQEC BBNVY BENPR BEZIV BGLVJ BHPHI CCPQU D1I DWQXO FRNLG F~G GNUQQ GUQSH HCIFZ JQ2 K60 K6~ K7- KB. L.- L6V LK8 M0C M0K M0N M1Q M2O M2P M2T M7P M7S MBDVC P5Z P62 PATMY PDBOC PHGZM PHGZT PKEHL PQBIZ PQBZA PQEST PQGLB PQQKQ PQUKI PRINS PTHSS PYCSY Q9U |
| DOI | 10.1137/090750020 |
| DatabaseName | CrossRef Pascal-Francis ProQuest Central (Corporate) ABI/INFORM Collection ABI/INFORM Global (PDF only) Agricultural Science Collection ProQuest Central (purchase pre-March 2016) ABI/INFORM Collection Biology Database (Alumni Edition) Military Database (Alumni Edition) Science Database (Alumni Edition) Telecommunications (Alumni Edition) Computing Database (Alumni Edition) ProQuest SciTech Collection ProQuest Technology Collection ProQuest Natural Science Collection ProQuest Central (Alumni) (purchase pre-March 2016) ABI/INFORM Collection (Alumni) Research Library (Alumni) ProQuest Materials Science & Engineering ProQuest Central (Alumni) ProQuest Central UK/Ireland Advanced Technologies & Computer Science Collection Agricultural & Environmental Science Collection ProQuest Central Essentials - QC Biological Science Collection ProQuest Central Business Premium Collection ProQuest Technology Collection Natural Science Collection ProQuest One Community College ProQuest Materials Science Collection ProQuest Central Business Premium Collection (Alumni) ABI/INFORM Global (Corporate) ProQuest Central Student ProQuest Research Library SciTech Premium Collection ProQuest Computer Science Collection ProQuest Business Collection (Alumni Edition) ProQuest Business Collection Computer Science Database Materials Science Database ABI/INFORM Professional Advanced ProQuest Engineering Collection Biological Sciences ABI/INFORM Global Agricultural Science Database Computing Database Military Database Research Library Science Database Telecommunications Database Biological Science Database Engineering Database Research Library (Corporate) Advanced Technologies & Aerospace Database ProQuest Advanced Technologies & Aerospace Collection Environmental Science Database Materials Science Collection Proquest Central Premium ProQuest One Academic (New) ProQuest One Academic Middle East (New) ProQuest One Business (UW System Shared) ProQuest One Business (Alumni) ProQuest One Academic Eastern Edition (DO NOT USE) One Applied & Life Sciences ProQuest One Academic (retired) ProQuest One Academic UKI Edition ProQuest Central China Engineering Collection Environmental Science Collection ProQuest Central Basic |
| DatabaseTitle | CrossRef Agricultural Science Database ProQuest Business Collection (Alumni Edition) Research Library Prep Computer Science Database ProQuest Central Student ProQuest Advanced Technologies & Aerospace Collection ProQuest Central Essentials ProQuest Computer Science Collection SciTech Premium Collection ProQuest Military Collection ProQuest Central China ABI/INFORM Complete ProQuest Telecommunications ProQuest One Applied & Life Sciences Natural Science Collection Biological Science Collection ProQuest Central (New) Engineering Collection Advanced Technologies & Aerospace Collection Business Premium Collection ABI/INFORM Global Engineering Database ProQuest Science Journals (Alumni Edition) ProQuest Biological Science Collection ProQuest One Academic Eastern Edition Agricultural Science Collection ProQuest Technology Collection ProQuest Telecommunications (Alumni Edition) Biological Science Database ProQuest Business Collection Environmental Science Collection ProQuest One Academic UKI Edition Environmental Science Database ProQuest One Academic ProQuest One Academic (New) ABI/INFORM Global (Corporate) ProQuest One Business Technology Collection ProQuest One Academic Middle East (New) Materials Science Collection ProQuest Central (Alumni Edition) ProQuest One Community College Research Library (Alumni Edition) ProQuest Natural Science Collection ProQuest Biology Journals (Alumni Edition) ProQuest Central ABI/INFORM Professional Advanced ProQuest Engineering Collection ProQuest Central Korea Agricultural & Environmental Science Collection Materials Science Database ProQuest Research Library ABI/INFORM Complete (Alumni Edition) ProQuest Materials Science Collection ProQuest Computing ABI/INFORM Global (Alumni Edition) ProQuest Central Basic ProQuest Science Journals ProQuest Computing (Alumni Edition) ProQuest Military Collection (Alumni Edition) ProQuest SciTech Collection Advanced Technologies & Aerospace Database Materials Science & Engineering Collection ProQuest One Business (Alumni) ProQuest Central (Alumni) Business Premium Collection (Alumni) |
| DatabaseTitleList | Agricultural Science Database |
| Database_xml | – sequence: 1 dbid: BENPR name: ProQuest Central url: https://www.proquest.com/central sourceTypes: Aggregation Database |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Computer Science Applied Sciences Mathematics |
| EISSN | 1095-7146 |
| EndPage | 2078 |
| ExternalDocumentID | 2599485641 23421152 10_1137_090750020 |
| GroupedDBID | .4S .DC 123 4.4 6TJ 7WY 7X2 7XC 88I 8CJ 8FE 8FG 8FH 8FL 8G5 AALVN AASXH AAYXX ABDBF ABDPE ABJCF ABKAD ABMZU ABUWG ACGFO ACGOD ACIWK ACPRK ACUHS ADBBV ADNWM ADXHL AENEX AFFHD AFFNX AFKRA AFRAH ALMA_UNASSIGNED_HOLDINGS ANXRF ARAPS ARCSS ATCPS AZQEC BBNVY BENPR BEZIV BGLVJ BHPHI BPHCQ CCPQU CITATION CS3 CZ9 D1I D1J D1K DQ2 DU5 DWQXO EAP EBS EDO EJD EMK EST ESX FRNLG GNUQQ GUQSH H13 HCIFZ H~9 I-F K6- K60 K6V K6~ K7- KB. KC. L6V LK5 LK8 M0C M0K M1Q M2O M2P M7P M7R M7S P1Q P2P P62 PATMY PDBOC PHGZM PHGZT PQBIZ PQBZA PQGLB PQQKQ PROAC PTHSS PYCSY RJG RNS RSI TN5 TUS UQL YNT YYP IQODW 3V. 7XB 88A 88K 8AL 8FK JQ2 L.- M0N M2T MBDVC PKEHL PQEST PQUKI PRINS Q9U |
| ID | FETCH-LOGICAL-c352t-f3c57e7327e19e6600c85bcd00181d5e2d2f9434ffbf6725ffd5e7588818c7ae3 |
| IEDL.DBID | K7- |
| ISICitedReferencesCount | 129 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000277834600005&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 0895-4801 |
| IngestDate | Sat Oct 25 10:03:23 EDT 2025 Mon Jul 21 09:12:00 EDT 2025 Sat Nov 29 05:37:00 EST 2025 Tue Nov 18 20:39:01 EST 2025 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 4 |
| Keywords | Hypergraph Constraint Combinatorial problem Matroid Optimization method Entropy 68W25 approximation algorithms Approximation algorithm Combinatorial optimization Location problem Maximum matroid and knapsack constraints Discrete mathematics Sampling Monotonic function Partition Approximation Maximization Minimization 90C27 Knapsack problem submodular maximization Graph cut Digraph Directed graph |
| Language | English |
| License | CC BY 4.0 |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c352t-f3c57e7327e19e6600c85bcd00181d5e2d2f9434ffbf6725ffd5e7588818c7ae3 |
| Notes | ObjectType-Article-1 SourceType-Scholarly Journals-1 content type line 14 |
| PQID | 925635158 |
| PQPubID | 666311 |
| PageCount | 26 |
| ParticipantIDs | proquest_journals_925635158 pascalfrancis_primary_23421152 crossref_primary_10_1137_090750020 crossref_citationtrail_10_1137_090750020 |
| PublicationCentury | 2000 |
| PublicationDate | 2010-01-01 |
| PublicationDateYYYYMMDD | 2010-01-01 |
| PublicationDate_xml | – month: 01 year: 2010 text: 2010-01-01 day: 01 |
| PublicationDecade | 2010 |
| PublicationPlace | Philadelphia, PA |
| PublicationPlace_xml | – name: Philadelphia, PA – name: Philadelphia |
| PublicationTitle | SIAM journal on discrete mathematics |
| PublicationYear | 2010 |
| Publisher | Society for Industrial and Applied Mathematics |
| Publisher_xml | – name: Society for Industrial and Applied Mathematics |
| References | R2 R3 R5 R6 Krause A. (R29) 2008; 9 R8 R9 R50 R30 R41 R40 R12 R23 R45 R33 R25 R24 R46 R16 R49 R18 R28 R19 R1 |
| References_xml | – ident: R1 doi: 10.1137/S089548010036813X – ident: R25 doi: 10.1145/502090.502096 – ident: R18 doi: 10.1145/227683.227684 – ident: R49 doi: 10.1080/02664768700000020 – ident: R33 doi: 10.1287/opre.46.5.655 – ident: R41 doi: 10.1007/BFb0121195 – ident: R2 doi: 10.1016/S0166-218X(99)00103-1 – ident: R23 doi: 10.1007/s00037-006-0205-6 – ident: R19 doi: 10.1287/mnsc.45.11.1539 – ident: R16 doi: 10.1016/0166-218X(83)90040-9 – ident: R50 doi: 10.1016/S0167-6377(03)00062-2 – ident: R30 doi: 10.1016/0377-2217(95)00205-7 – ident: R24 doi: 10.1145/502090.502098 – volume: 9 start-page: 235 issn: 1532-4435 year: 2008 ident: R29 publication-title: J. Mach. Learn. Res. – ident: R40 doi: 10.1007/BF01588971 – ident: R5 doi: 10.1007/s101070050055 – ident: R3 doi: 10.1023/B:JOCO.0000038913.96607.c2 – ident: R45 doi: 10.1137/0910022 – ident: R46 doi: 10.1006/jctb.2000.1989 – ident: R6 doi: 10.1007/s10107-006-0024-1 – ident: R8 doi: 10.1287/mnsc.23.8.789 – ident: R12 doi: 10.1145/285055.285059 – ident: R28 doi: 10.1287/opre.43.4.684 – ident: R9 doi: 10.1016/S0167-5060(08)70732-5 |
| SSID | ssj0014415 |
| Score | 2.2552927 |
| Snippet | Submodular function maximization is a central problem in combinatorial optimization, generalizing many important problems including Max Cut in... |
| SourceID | proquest pascalfrancis crossref |
| SourceType | Aggregation Database Index Database Enrichment Source |
| StartPage | 2053 |
| SubjectTerms | Algebra Algorithmics. Computability. Computer arithmetics Algorithms Applied mathematics Applied sciences Approximation Combinatorics Combinatorics. Ordered structures Computer science; control theory; systems Convex and discrete geometry Design of experiments Entropy Exact sciences and technology Geometry Graph theory Graphs Mathematical programming Mathematics Optimization Random variables Sciences and techniques of general use Theoretical computing |
| Title | Maximizing Nonmonotone Submodular Functions under Matroid or Knapsack Constraints |
| URI | https://www.proquest.com/docview/925635158 |
| Volume | 23 |
| WOSCitedRecordID | wos000277834600005&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: PRVPQU databaseName: ABI/INFORM Global customDbUrl: eissn: 1095-7146 dateEnd: 20140731 omitProxy: false ssIdentifier: ssj0014415 issn: 0895-4801 databaseCode: M0C dateStart: 19880201 isFulltext: true titleUrlDefault: https://search.proquest.com/abiglobal providerName: ProQuest – providerCode: PRVPQU databaseName: Agricultural Science Database customDbUrl: eissn: 1095-7146 dateEnd: 20140731 omitProxy: false ssIdentifier: ssj0014415 issn: 0895-4801 databaseCode: M0K dateStart: 19880201 isFulltext: true titleUrlDefault: https://search.proquest.com/agriculturejournals providerName: ProQuest – providerCode: PRVPQU databaseName: Biological Science Database customDbUrl: eissn: 1095-7146 dateEnd: 20140731 omitProxy: false ssIdentifier: ssj0014415 issn: 0895-4801 databaseCode: M7P dateStart: 19880201 isFulltext: true titleUrlDefault: http://search.proquest.com/biologicalscijournals providerName: ProQuest – providerCode: PRVPQU databaseName: Computer Science Database customDbUrl: eissn: 1095-7146 dateEnd: 20140731 omitProxy: false ssIdentifier: ssj0014415 issn: 0895-4801 databaseCode: K7- dateStart: 19880201 isFulltext: true titleUrlDefault: http://search.proquest.com/compscijour providerName: ProQuest – providerCode: PRVPQU databaseName: Engineering Database customDbUrl: eissn: 1095-7146 dateEnd: 20140731 omitProxy: false ssIdentifier: ssj0014415 issn: 0895-4801 databaseCode: M7S dateStart: 19880201 isFulltext: true titleUrlDefault: http://search.proquest.com providerName: ProQuest – providerCode: PRVPQU databaseName: Environmental Science Database customDbUrl: eissn: 1095-7146 dateEnd: 20140731 omitProxy: false ssIdentifier: ssj0014415 issn: 0895-4801 databaseCode: PATMY dateStart: 19880201 isFulltext: true titleUrlDefault: http://search.proquest.com/environmentalscience providerName: ProQuest – providerCode: PRVPQU databaseName: Materials Science Database customDbUrl: eissn: 1095-7146 dateEnd: 20140731 omitProxy: false ssIdentifier: ssj0014415 issn: 0895-4801 databaseCode: KB. dateStart: 19880201 isFulltext: true titleUrlDefault: http://search.proquest.com/materialsscijournals providerName: ProQuest – providerCode: PRVPQU databaseName: Military Database customDbUrl: eissn: 1095-7146 dateEnd: 20140731 omitProxy: false ssIdentifier: ssj0014415 issn: 0895-4801 databaseCode: M1Q dateStart: 19880201 isFulltext: true titleUrlDefault: https://search.proquest.com/military providerName: ProQuest – providerCode: PRVPQU databaseName: ProQuest ABI/INFORM Collection customDbUrl: eissn: 1095-7146 dateEnd: 20140731 omitProxy: false ssIdentifier: ssj0014415 issn: 0895-4801 databaseCode: 7WY dateStart: 19880201 isFulltext: true titleUrlDefault: https://www.proquest.com/abicomplete providerName: ProQuest – providerCode: PRVPQU databaseName: ProQuest advanced technologies & aerospace journals customDbUrl: eissn: 1095-7146 dateEnd: 20140731 omitProxy: false ssIdentifier: ssj0014415 issn: 0895-4801 databaseCode: P5Z dateStart: 19880201 isFulltext: true titleUrlDefault: https://search.proquest.com/hightechjournals providerName: ProQuest – providerCode: PRVPQU databaseName: ProQuest Central customDbUrl: eissn: 1095-7146 dateEnd: 20140731 omitProxy: false ssIdentifier: ssj0014415 issn: 0895-4801 databaseCode: BENPR dateStart: 19880201 isFulltext: true titleUrlDefault: https://www.proquest.com/central providerName: ProQuest – providerCode: PRVPQU databaseName: ProQuest research library customDbUrl: eissn: 1095-7146 dateEnd: 20140731 omitProxy: false ssIdentifier: ssj0014415 issn: 0895-4801 databaseCode: M2O dateStart: 19880201 isFulltext: true titleUrlDefault: https://search.proquest.com/pqrl providerName: ProQuest – providerCode: PRVPQU databaseName: Science Database customDbUrl: eissn: 1095-7146 dateEnd: 20140731 omitProxy: false ssIdentifier: ssj0014415 issn: 0895-4801 databaseCode: M2P dateStart: 19880201 isFulltext: true titleUrlDefault: https://search.proquest.com/sciencejournals providerName: ProQuest |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV1Lj9MwEB6xuxyQEMtTlIXKQhy4GGonjpMTYldbIVUtXVjEwiVy_JAq2KY0BSF-PTOuG4FAXLh8ihJHsTz2POzJfABPAgbUuTOCm1J4nmtveaPpuDAvCmcMOtzKRLIJPZuVFxfVPOXmdCmtcqcTo6J2raU98ucV2uYMjW_5YvWFE2kUHa4mBo09OBBSCprmE837QwQKFaITWSlOVVJSYSFBJX0qMpUjYvn-xRxdX5kORyZsKS3-0M7R5IwP_7OzN-FG8jXZy-3kuAVX_PI2HO54HFha1nfgbGq-Ly4XP9CKsVm7xLdbKtHNUKlcto7yVNkYzV-coYx-OluzKW2hLxxr12yyNKvO2E-MuD8j48SmuwvvxqfnJ694olrgFj2wDQ-ZVdrrTGovKl-gF2RL1VhHnH3CKS-dDFRJLoQmFFqqEPAmhhol2nurjc_uwf4Se3Yf2KhRRTBK-KbEUBOVWBDWlVkuA7pKNm8G8HQ34rVNdcipc5_rGI9kuu6FM4DHfdPVtvjG3xoNfxNb31LiN9HflQM42gmpTgu0q3sJPfjn0yO4tk0XoD2Xh7C_WX_1j-Cq_bZZdOsh7On3H4ZwcHw6m78ZxslHePwMcTo6iTghFGeE8nXEecRzQh2v9VvEufr4E1NH6W0 |
| linkProvider | ProQuest |
| linkToHtml | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMw1V1bS9xAFD5stdBCUWtbXFftIC30JbiZXCZ5EBF1UdZdWrDUPsXJXGCpbrabbb38J_-j5-RGRfHNh74mQzKZ-XK-c-bMnA_gk8WA2tfSdWTkGscXRjmpoHShH4ZaSnS4A1mITYjhMDo9jb-24LY-C0PbKmubWBhqnSlaI9-KkZs9JN9oZ_LbIdEoSq7WCholKvrm-hIjtnz7aB-n9zPnvYOTvUOnEhVwFPoaM8d6KhBGeFwYNzYh8r2KglRpUqdzdWC45pZqplmb2lDwwFq8iE51hMymhDQePvcFzKMZF6QUIX78bJIWFJoUTmscOFSVpSpk5FIJoZiouUuq4v_Q35uJzHEmbCmh8YANCorrLf5ng7MEC5UvzXZL8L-Flhkvw2KtU8Eqs_UOvg3k1ehidIMszYbZGHubUQlyhkbzItO0D5f1kN6LP5DRobopG1CKYKRZNmX9sZzkUv1ipG1aKGrM8vfw_Vk-7APMjbFnK8C6aRBaGbgmjTCURiNtXaUjz-cWXUHlp234Us9woqo669S586SItzyRNGBow2bTdFIWF3ms0cY9mDQtOb4T_Xnehk4NiqQyQHnSIGL1ybsf4dXhyeA4OT4a9jvwutwaQetLazA3m_4x6_BS_Z2N8ulGAXUGZ88NnzvWRDrr |
| 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=Maximizing+Nonmonotone+Submodular+Functions+under+Matroid+or+Knapsack+Constraints&rft.jtitle=SIAM+journal+on+discrete+mathematics&rft.au=Lee%2C+Jon&rft.au=Mirrokni%2C+Vahab+S&rft.au=Nagarajan%2C+Viswanath&rft.au=Sviridenko%2C+Maxim&rft.date=2010-01-01&rft.pub=Society+for+Industrial+and+Applied+Mathematics&rft.issn=0895-4801&rft.eissn=1095-7146&rft.volume=23&rft.issue=4&rft.spage=2053&rft_id=info:doi/10.1137%2F090750020&rft.externalDocID=2599485641 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0895-4801&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0895-4801&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0895-4801&client=summon |