A Unified Continuous Greedy Algorithm for Submodular Maximization

The study of combinatorial problems with a submodular objective function has attracted much attention in recent years, and is partly motivated by the importance of such problems to economics, algorithmic game theory and combinatorial optimization. Classical works on these problems are mostly combina...

Full description

Saved in:
Bibliographic Details
Published in:2011 IEEE 52nd Annual Symposium on Foundations of Computer Science pp. 570 - 579
Main Authors: Feldman, Moran, Naor, Joseph, Schwartz, Roy
Format: Conference Proceeding
Language:English
Published: IEEE 01.10.2011
Subjects:
Algorithm design and analysis
Approximation
Approximation algorithms
Approximation methods
Continuous Greedy
Greedy algorithms
Max-SAT
Optimized production technology
Submodular
Submodular Welfare
Vectors
ISBN:145771843X, 9781457718434
ISSN:0272-5428
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Abstract The study of combinatorial problems with a submodular objective function has attracted much attention in recent years, and is partly motivated by the importance of such problems to economics, algorithmic game theory and combinatorial optimization. Classical works on these problems are mostly combinatorial in nature. Recently, however, many results based on continuous algorithmic tools have emerged. The main bottleneck of such continuous techniques is how to approximately solve a non-convex relaxation for the sub- modular problem at hand. Thus, the efficient computation of better fractional solutions immediately implies improved approximations for numerous applications. A simple and elegant method, called "continuous greedy", successfully tackles this issue for monotone submodular objective functions, however, only much more complex tools are known to work for general non-monotone submodular objectives. In this work we present a new unified continuous greedy algorithm which finds approximate fractional solutions for both the non-monotone and monotone cases, and improves on the approximation ratio for many applications. For general non-monotone submodular objective functions, our algorithm achieves an improved approximation ratio of about 1/e. For monotone submodular objective functions, our algorithm achieves an approximation ratio that depends on the density of the polytope defined by the problem at hand, which is always at least as good as the previously known best approximation ratio of 1-1/e. Some notable immediate implications are an improved 1/e-approximation for maximizing a non-monotone submodular function subject to a matroid or O(1)-knapsack constraints, and information-theoretic tight approximations for Submodular Max-SAT and Submodular Welfare with k players, for any number of players k. A framework for submodular optimization problems, called the contention resolution framework, was introduced recently by Chekuri et al. [11]. The improved approximation ratio of the unified continuous greedy algorithm implies improved ap- proximation ratios for many problems through this framework. Moreover, via a parameter called stopping time, our algorithm merges the relaxation solving and re-normalization steps of the framework, and achieves, for some applications, further improvements. We also describe new monotone balanced con- tention resolution schemes for various matching, scheduling and packing problems, thus, improving the approximations achieved for these problems via the framework.
AbstractList The study of combinatorial problems with a submodular objective function has attracted much attention in recent years, and is partly motivated by the importance of such problems to economics, algorithmic game theory and combinatorial optimization. Classical works on these problems are mostly combinatorial in nature. Recently, however, many results based on continuous algorithmic tools have emerged. The main bottleneck of such continuous techniques is how to approximately solve a non-convex relaxation for the sub- modular problem at hand. Thus, the efficient computation of better fractional solutions immediately implies improved approximations for numerous applications. A simple and elegant method, called "continuous greedy", successfully tackles this issue for monotone submodular objective functions, however, only much more complex tools are known to work for general non-monotone submodular objectives. In this work we present a new unified continuous greedy algorithm which finds approximate fractional solutions for both the non-monotone and monotone cases, and improves on the approximation ratio for many applications. For general non-monotone submodular objective functions, our algorithm achieves an improved approximation ratio of about 1/e. For monotone submodular objective functions, our algorithm achieves an approximation ratio that depends on the density of the polytope defined by the problem at hand, which is always at least as good as the previously known best approximation ratio of 1-1/e. Some notable immediate implications are an improved 1/e-approximation for maximizing a non-monotone submodular function subject to a matroid or O(1)-knapsack constraints, and information-theoretic tight approximations for Submodular Max-SAT and Submodular Welfare with k players, for any number of players k. A framework for submodular optimization problems, called the contention resolution framework, was introduced recently by Chekuri et al. [11]. The improved approximation ratio of the unified continuous greedy algorithm implies improved ap- proximation ratios for many problems through this framework. Moreover, via a parameter called stopping time, our algorithm merges the relaxation solving and re-normalization steps of the framework, and achieves, for some applications, further improvements. We also describe new monotone balanced con- tention resolution schemes for various matching, scheduling and packing problems, thus, improving the approximations achieved for these problems via the framework.
Author Naor, Joseph
Schwartz, Roy
Feldman, Moran
Author_xml – sequence: 1
  givenname: Moran
  surname: Feldman
  fullname: Feldman, Moran
  email: moranfe@cs.technion.ac.il
  organization: Dept. of Comput. Sci., Technion - Israel Inst. of Technol., Haifa, Israel
– sequence: 2
  givenname: Joseph
  surname: Naor
  fullname: Naor, Joseph
  email: naor@cs.technion.ac.il
  organization: Dept. of Comput. Sci., Technion - Israel Inst. of Technol., Haifa, Israel
– sequence: 3
  givenname: Roy
  surname: Schwartz
  fullname: Schwartz, Roy
  email: schwartz@cs.technion.ac.il
  organization: Dept. of Comput. Sci., Technion - Israel Inst. of Technol., Haifa, Israel
BookMark eNotzM1KAzEUQOGIFWxrd-7c5AWm5iaZ5M5yGGwVKl3UgrsyMTca6ExkfsD69C3o6nyrM2OTNrXE2D2IJYAoHlfbareUAmCpzRWbCWuKXOcW8JrN4IKLtHqfsKmQVma5lnjLFn0fnYAclTRCT1lZ8n0bQyTPq9QOsR3T2PN1R-RPvDx-pi4OXw0PqeO70TXJj8e646_1T2zibz3E1N6xm1Afe1r8d872q6e36jnbbNcvVbnJotQwZEjolfCOSEFADCEUlCsAp5113nxIwiBISdQyCOOp1mRtQIMFWXK5UnP28PeNRHT47mJTd6eDAYESUJ0BzsZN7A
ContentType Conference Proceeding
DBID 6IE
6IH
CBEJK
RIE
RIO
DOI 10.1109/FOCS.2011.46
DatabaseName IEEE Electronic Library (IEL) Conference Proceedings
IEEE Proceedings Order Plan (POP) 1998-present by volume
IEEE Xplore All Conference Proceedings
IEEE Electronic Library (IEL)
IEEE Proceedings Order Plans (POP) 1998-present
DatabaseTitleList
Database_xml – sequence: 1
  dbid: RIE
  name: IEEE Xplore
  url: https://ieeexplore.ieee.org/
  sourceTypes: Publisher
DeliveryMethod fulltext_linktorsrc
Discipline Mathematics
EISBN 0769545718
9780769545714
EndPage 579
ExternalDocumentID 6108218
Genre orig-research
GroupedDBID -~X
23M
6IE
6IF
6IH
6IK
6IL
6IM
6IN
AAJGR
AAWTH
ACNCT
ADZIZ
AFFNX
AI.
ALMA_UNASSIGNED_HOLDINGS
BEFXN
BFFAM
BGNUA
BKEBE
BPEOZ
CBEJK
CHZPO
IPLJI
OCL
RIE
RIL
RIO
RNS
VH1
ID FETCH-LOGICAL-i241t-8e8d30dbee31f88fff9e5311b4b7bd6c2e8f0e32842f06dea4e77f8689e7eb533
IEDL.DBID RIE
ISBN 145771843X
9781457718434
ISICitedReferencesCount 187
ISICitedReferencesURI http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000298962700061&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
ISSN 0272-5428
IngestDate Wed Aug 27 02:47:58 EDT 2025
IsPeerReviewed false
IsScholarly false
Language English
LinkModel DirectLink
MergedId FETCHMERGED-LOGICAL-i241t-8e8d30dbee31f88fff9e5311b4b7bd6c2e8f0e32842f06dea4e77f8689e7eb533
PageCount 10
ParticipantIDs ieee_primary_6108218
PublicationCentury 2000
PublicationDate 2011-10
PublicationDateYYYYMMDD 2011-10-01
PublicationDate_xml – month: 10
  year: 2011
  text: 2011-10
PublicationDecade 2010
PublicationTitle 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
PublicationTitleAbbrev focs
PublicationYear 2011
Publisher IEEE
Publisher_xml – name: IEEE
SSID ssib015832604
ssj0040503
ssj0000669372
Score 1.9550357
Snippet The study of combinatorial problems with a submodular objective function has attracted much attention in recent years, and is partly motivated by the...
SourceID ieee
SourceType Publisher
StartPage 570
SubjectTerms Algorithm design and analysis
Approximation
Approximation algorithms
Approximation methods
Continuous Greedy
Greedy algorithms
Max-SAT
Optimized production technology
Submodular
Submodular Welfare
Vectors
Title A Unified Continuous Greedy Algorithm for Submodular Maximization
URI https://ieeexplore.ieee.org/document/6108218
WOSCitedRecordID wos000298962700061&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
link http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV27TsMwFLXaigEWHi3iLQ-MhObh-DFWFRVLSyVA6lbF9jVEoglqEwR_j500gYGFLc5k3Wv7HNnnnovQdagSmkCkPSWi2CMyNnbPRYlHfSVk7BMTqaRqNsFmM75YiHkH3bS1MABQic_g1n1Wb_k6V6W7KhtaqOcWkrqoyxita7WatRPEdmk2PiP1KUwt8rZWUsT5nlT3Lcx5_4e8KvKKGXP9ThaN99N2TFqFvBhOHsaPtdOn48i_OrBUADTZ_9_UD9Dgp5IPz1uMOkQdyI7Q3rT1at300WiELfE0lopiZ1WVZmVebrDT4-gvPHp7yddp8brCltxie8qscu2Eq3iafKarbQ3nAD1P7p7G9962sYKXWsAuPA5cR76WAFFgODfGCLB7MZBEMqmpCoEbHyKLXKHxqYaEAGOGUy6AgbQE8Rj1sjyDE4RNokIKJlaCGGJ8ZZMttY22YpFNuaCnqO-CsXyvvTOW2zic_f37HO2GjcYuuEC9Yl3CJdpRH0W6WV9VCf8Grcyk0A
linkProvider IEEE
linkToHtml http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV09T8MwELVKQQIWPlrENx4YCU1iJ3bGqqIqoi2VKFK3KrHPEIkmqE0Q_HvspAkMLGxxJuvO9nuy371D6NoVoR8CkZYIiGfRyFN6z5HQ8m0RRJ5NFRFh0WyCjcd8NgsmDXRT18IAQCE-g1vzWbzly1Tk5qqso6Gea0jaQJsepa5dVmtVq8fx9OKsnEbKc9jX2FubSVHjfFLcuDDj_u_yoszLY8x0PJlV7k_rMa018kGn_9h7Kr0-DUv-1YOlgKD-3v8mv4_aP7V8eFKj1AFqQHKIdke1W-uqhbpdrKmn0mQUG7OqOMnTfIWNIkd-4e7bS7qMs9cF1vQW63NmkUojXcWj8DNerKs42-i5fzftDax1awUr1pCdWRy4JLaMAIijOFdKBaB3oxPRiEXSFy5wZQPR2OUq25cQUmBMcZ8HwCDSFPEINZM0gWOEVShcH5QnAqqosoVOdyR1tAUjOumBf4JaJhjz99I9Y76Ow-nfv6_Q9mA6Gs6H9-OHM7TjVoo75xw1s2UOF2hLfGTxanlZJP8bd9yoFw
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%3Abook&rft.genre=proceeding&rft.title=2011+IEEE+52nd+Annual+Symposium+on+Foundations+of+Computer+Science&rft.atitle=A+Unified+Continuous+Greedy+Algorithm+for+Submodular+Maximization&rft.au=Feldman%2C+Moran&rft.au=Naor%2C+Joseph&rft.au=Schwartz%2C+Roy&rft.date=2011-10-01&rft.pub=IEEE&rft.isbn=9781457718434&rft.issn=0272-5428&rft.spage=570&rft.epage=579&rft_id=info:doi/10.1109%2FFOCS.2011.46&rft.externalDocID=6108218
thumbnail_l http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0272-5428&client=summon
thumbnail_m http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0272-5428&client=summon
thumbnail_s http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0272-5428&client=summon