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Title: Submodular optimization problems and greedy strategies: A survey

Abstract

The greedy strategy is an approximation algorithm to solve optimization problems arising in decision making with multiple actions. How effective is the greedy strategy compared to the optimal solution? In this survey, we mainly consider two classes of optimization problems where the objective function is submodular. The first is set submodular optimization, which is to choose a set of actions to optimize a set submodular objective function, and the second is string submodular optimization, which is to choose an ordered set of actions to optimize a string submodular function. Our emphasis here is on performance bounds for the greedy strategy in submodular optimization problems. Specifically, we review performance bounds for the greedy strategy, more general and improved bounds in terms of curvature, performance bounds for the batched greedy strategy, and performance bounds for Nash equilibria.

Authors:
ORCiD logo [1];  [2];  [2];  [3]
  1. National Renewable Energy Lab. (NREL), Golden, CO (United States)
  2. Colorado State Univ., Fort Collins, CO (United States)
  3. Alibaba iDST, Seattle, WA (United States)
Publication Date:
Research Org.:
National Renewable Energy Lab. (NREL), Golden, CO (United States)
Sponsoring Org.:
USDOE Office of Energy Efficiency and Renewable Energy (EERE)
OSTI Identifier:
1603264
Report Number(s):
[NREL/JA-5D00-72380]
[Journal ID: ISSN 0924-6703]
Grant/Contract Number:  
[AC36-08GO28308]
Resource Type:
Accepted Manuscript
Journal Name:
Discrete Event Dynamic Systems
Additional Journal Information:
[Journal Name: Discrete Event Dynamic Systems]; Journal ID: ISSN 0924-6703
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; curvature; greedy strategy; Nash equilibrium; performance; optimization; submodular

Citation Formats

Liu, Yajing, Chong, Edwin K. P., Pezeshki, Ali, and Zhang, Zhenliang. Submodular optimization problems and greedy strategies: A survey. United States: N. p., 2020. Web. doi:10.1007/s10626-019-00308-7.
Liu, Yajing, Chong, Edwin K. P., Pezeshki, Ali, & Zhang, Zhenliang. Submodular optimization problems and greedy strategies: A survey. United States. doi:10.1007/s10626-019-00308-7.
Liu, Yajing, Chong, Edwin K. P., Pezeshki, Ali, and Zhang, Zhenliang. Mon . "Submodular optimization problems and greedy strategies: A survey". United States. doi:10.1007/s10626-019-00308-7.
@article{osti_1603264,
title = {Submodular optimization problems and greedy strategies: A survey},
author = {Liu, Yajing and Chong, Edwin K. P. and Pezeshki, Ali and Zhang, Zhenliang},
abstractNote = {The greedy strategy is an approximation algorithm to solve optimization problems arising in decision making with multiple actions. How effective is the greedy strategy compared to the optimal solution? In this survey, we mainly consider two classes of optimization problems where the objective function is submodular. The first is set submodular optimization, which is to choose a set of actions to optimize a set submodular objective function, and the second is string submodular optimization, which is to choose an ordered set of actions to optimize a string submodular function. Our emphasis here is on performance bounds for the greedy strategy in submodular optimization problems. Specifically, we review performance bounds for the greedy strategy, more general and improved bounds in terms of curvature, performance bounds for the batched greedy strategy, and performance bounds for Nash equilibria.},
doi = {10.1007/s10626-019-00308-7},
journal = {Discrete Event Dynamic Systems},
number = ,
volume = ,
place = {United States},
year = {2020},
month = {2}
}

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Works referenced in this record:

Optimization Strategies in Adaptive Control: A Selective Survey
journal, January 1975


Approximation for maximizing monotone non-decreasing set functions with a greedy method
journal, January 2014

  • Wang, Zengfu; Moran, Bill; Wang, Xuezhi
  • Journal of Combinatorial Optimization, Vol. 31, Issue 1
  • DOI: 10.1007/s10878-014-9707-3

Alternative Distributed Algorithms for Network Utility Maximization: Framework and Applications
journal, December 2007

  • Palomar, Daniel P.; Chiang, Mung
  • IEEE Transactions on Automatic Control, Vol. 52, Issue 12
  • DOI: 10.1109/TAC.2007.910665

The maximal covering location problem
journal, December 1974

  • Church, Richard; ReVelle, Charles
  • Papers of the Regional Science Association, Vol. 32, Issue 1
  • DOI: 10.1007/BF01942293

Maximizing a Monotone Submodular Function Subject to a Matroid Constraint
journal, January 2011

  • Calinescu, Gruia; Chekuri, Chandra; Pál, Martin
  • SIAM Journal on Computing, Vol. 40, Issue 6
  • DOI: 10.1137/080733991

Exceptional Paper—Location of Bank Accounts to Optimize Float: An Analytic Study of Exact and Approximate Algorithms
journal, April 1977

  • Cornuejols, Gerard; Fisher, Marshall L.; Nemhauser, George L.
  • Management Science, Vol. 23, Issue 8
  • DOI: 10.1287/mnsc.23.8.789

An inequality for polymatroid functions and its applications
journal, September 2003


Maximizing Submodular Set Functions Subject to Multiple Linear Constraints
conference, December 2013

  • Kulik, Ariel; Shachnai, Hadas; Tamir, Tami
  • Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
  • DOI: 10.1137/1.9781611973068.60

A note on maximizing a submodular set function subject to a knapsack constraint
journal, January 2004


Performance bounds for the k-batch greedy strategy in optimization problems with curvature
conference, July 2016

  • Liu, Yajing; Zhang, Zhenliang; Chong, Edwin K. P.
  • 2016 American Control Conference (ACC)
  • DOI: 10.1109/ACC.2016.7526805

Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming
journal, November 1995

  • Goemans, Michel X.; Williamson, David P.
  • Journal of the ACM, Vol. 42, Issue 6
  • DOI: 10.1145/227683.227684

Performance bounds for Nash equilibria in submodular utility systems with user groups
journal, October 2017


Optimal approximation for the submodular welfare problem in the value oracle model
conference, January 2008

  • Vondrak, Jan
  • the 40th annual ACM symposium, Proceedings of the fourtieth annual ACM symposium on Theory of computing - STOC 08
  • DOI: 10.1145/1374376.1374389

A submodularity-based approach for multi-agent optimal coverage problems
conference, December 2017

  • Sun, Xinmiao; Cassandras, Christos G.; Meng, Xiangyu
  • 2017 IEEE 56th Annual Conference on Decision and Control (CDC)
  • DOI: 10.1109/CDC.2017.8264258

Streaming submodular maximization: massive data summarization on the fly
conference, January 2014

  • Badanidiyuru, Ashwinkumar; Mirzasoleiman, Baharan; Karbasi, Amin
  • Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining - KDD '14
  • DOI: 10.1145/2623330.2623637

An analysis of approximations for maximizing submodular set functions—I
journal, December 1978

  • Nemhauser, G. L.; Wolsey, L. A.; Fisher, M. L.
  • Mathematical Programming, Vol. 14, Issue 1
  • DOI: 10.1007/BF01588971

Solving the Generalized Assignment Problem: An Optimizing and Heuristic Approach
journal, August 2003


Online submodular welfare maximization: Greedy is optimal
conference, December 2013

  • Kapralov, Michael; Post, Ian; Vondrák, Jan
  • Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms
  • DOI: 10.1137/1.9781611973105.88

Tight approximation algorithms for maximum general assignment problems
conference, January 2006

  • Fleischer, Lisa; Goemans, Michel X.; Mirrokni, Vahab S.
  • Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm - SODA '06
  • DOI: 10.1145/1109557.1109624

Extending Polymatroid Set Functions With Curvature and Bounding the Greedy Strategy
conference, June 2018

  • Liu, Yajing; Chong, Edwin K. P.; Pezeshki, Ali
  • 2018 IEEE Statistical Signal Processing Workshop (SSP)
  • DOI: 10.1109/SSP.2018.8450732

A survey of sensor selection schemes in wireless sensor networks
conference, May 2007

  • Rowaihy, Hosam; Eswaran, Sharanya; Johnson, Matthew
  • Defense and Security Symposium, SPIE Proceedings
  • DOI: 10.1117/12.723514

Submodular Maximization with Cardinality Constraints
conference, January 2014

  • Buchbinder, Niv; Feldman, Moran; Naor, Joseph (Seffi)
  • Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms
  • DOI: 10.1137/1.9781611973402.106

A Unified Continuous Greedy Algorithm for Submodular Maximization
conference, October 2011

  • Feldman, Moran; Naor, Joseph; Schwartz, Roy
  • 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS 2011)
  • DOI: 10.1109/FOCS.2011.46

An efficient approximation for the Generalized Assignment Problem
journal, November 2006


Submodular set functions, matroids and the greedy algorithm: Tight worst-case bounds and some generalizations of the Rado-Edmonds theorem
journal, March 1984


Greedy Adaptive Linear Compression in Signal-Plus-Noise Models
journal, April 2014

  • Liu, Entao; Chong, Edwin K. P.; Scharf, Louis L.
  • IEEE Transactions on Information Theory, Vol. 60, Issue 4
  • DOI: 10.1109/TIT.2014.2308258

Dynamic optimization using adaptive control vector parameterization
journal, July 2005


Approximate stochastic dynamic programming for sensor scheduling to track multiple targets
journal, December 2009


Towards Robust Multi-Layer Traffic Engineering: Optimization of Congestion Control and Routing
journal, June 2007

  • He, Jiayue; Bresler, Ma'ayan; Chiang, Mung
  • IEEE Journal on Selected Areas in Communications, Vol. 25, Issue 5
  • DOI: 10.1109/JSAC.2007.070602

An approximation algorithm for the generalized assignment problem
journal, February 1993

  • Shmoys, David B.; Tardos, Éva
  • Mathematical Programming, Vol. 62, Issue 1-3
  • DOI: 10.1007/BF01585178

Online Submodular Welfare Maximization: Greedy Beats 1/2 in Random Order
conference, January 2015

  • Korula, Nitish; Mirrokni, Vahab; Zadimoghaddam, Morteza
  • Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing - STOC '15
  • DOI: 10.1145/2746539.2746626

Transversals and matroid partition
journal, July 1965

  • Edmonds, J.; Fulkerso, D. R.
  • Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics, Vol. 69B, Issue 3
  • DOI: 10.6028/jres.069B.016

Improved bounds for the greedy strategy in optimization problems with curvature
journal, August 2018

  • Liu, Yajing; Chong, Edwin K. P.; Pezeshki, Ali
  • Journal of Combinatorial Optimization, Vol. 37, Issue 4
  • DOI: 10.1007/s10878-018-0345-z

Non-Cooperative Games
journal, September 1951

  • Nash, John
  • The Annals of Mathematics, Vol. 54, Issue 2
  • DOI: 10.2307/1969529

The budgeted maximum coverage problem
journal, April 1999


P-Complete Approximation Problems
journal, July 1976

  • Sahni, Sartaj; Gonzalez, Teofilo
  • Journal of the ACM (JACM), Vol. 23, Issue 3
  • DOI: 10.1145/321958.321975

Bounding the greedy strategy in finite-horizon string optimization
conference, December 2015

  • Liu, Yajing; Chong, Edwin K. P.; Pezeshki, Ali
  • 2015 54th IEEE Conference on Decision and Control (CDC)
  • DOI: 10.1109/CDC.2015.7402826

Utility-based rate control in the Internet for elastic traffic
journal, April 2002

  • La, R. J.; Anantharam, V.
  • IEEE/ACM Transactions on Networking, Vol. 10, Issue 2
  • DOI: 10.1109/90.993307

A Survey of Multi-Objective Sequential Decision-Making
journal, October 2013

  • Roijers, D. M.; Vamplew, P.; Whiteson, S.
  • Journal of Artificial Intelligence Research, Vol. 48
  • DOI: 10.1613/jair.3987

Nash equilibria in competitive societies, with applications to facility location, traffic routing and auctions
conference, January 2002

  • Vetta, A.
  • The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings.
  • DOI: 10.1109/SFCS.2002.1181966

Maximising Real-Valued Submodular Functions: Primal and Dual Heuristics for Location Problems
journal, August 1982


Performance Bounds with Curvature for Batched Greedy Optimization
journal, March 2018

  • Liu, Yajing; Zhang, Zhenliang; Chong, Edwin K. P.
  • Journal of Optimization Theory and Applications, Vol. 177, Issue 2
  • DOI: 10.1007/s10957-017-1177-1

Maximizing a class of submodular utility functions
journal, August 2009


Autonomous Vehicle-Target Assignment: A Game-Theoretical Formulation
journal, April 2007

  • Arslan, Gürdal; Marden, Jason R.; Shamma, Jeff S.
  • Journal of Dynamic Systems, Measurement, and Control, Vol. 129, Issue 5
  • DOI: 10.1115/1.2766722

Improved Bounds for Matroid Partition and Intersection Algorithms
journal, November 1986

  • Cunningham, William H.
  • SIAM Journal on Computing, Vol. 15, Issue 4
  • DOI: 10.1137/0215066

String Submodular Functions With Curvature Constraints
journal, March 2016

  • Zhang, Zhenliang; Chong, Edwin K. P.; Pezeshki, Ali
  • IEEE Transactions on Automatic Control, Vol. 61, Issue 3
  • DOI: 10.1109/TAC.2015.2440566

A submodular optimization framework for leader selection in linear multi-agent systems
conference, December 2011

  • Clark, Andrew; Poovendran, Radha
  • 2011 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC 2011)
  • DOI: 10.1109/CDC.2011.6160248