American Institute of Mathematical Sciences

Advanced
2015, 5(2): 91-100. doi: 10.3934/naco.2015.5.91

Primal-dual approximation algorithms for submodular cost set cover problems with linear/submodular penalties

Fengmin Wang 1, , Dachuan Xu 1, , Donglei Du 2, and  Chenchen Wu 3,

1. 

Department of Information and Operations Research, College of Applied Sciences, Beijing University of Technology, 100 Pingleyuan, Chaoyang District, Beijing 100124, China, China
2. 

Faculty of Business Administration, University of New Brunswick, Fredericton, NB, E3B 5A3, Canada
3. 

College of Science, Tianjin University of Technology, Tianjin 300384

Received  October 2014 Revised  April 2015 Published  June 2015

We introduce two set cover problems with submodular costs and linear/submodular penalties and offer two approximation algorithms of ratios $\eta$ and $2\eta$ respectively via the primal-dual technique, where $\eta$ is the largest number of sets that each element belongs to.
Keywords: primal-dual., penalties, submodular cost, Set cover, approximation algorithm.
Mathematics Subject Classification: Primary: 90C27; Secondary: 90C5.
Citation: Fengmin Wang, Dachuan Xu, Donglei Du, Chenchen Wu. Primal-dual approximation algorithms for submodular cost set cover problems with linear/submodular penalties. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 91-100. doi: 10.3934/naco.2015.5.91
References:
[1]

E. Balas, The prize collecting traveling salesman problem, Networks, 19 (1989), 621-636. doi: 10.1002/net.3230190602.  Google Scholar

[2]

D. Bienstock, M. Goemans, D. Simchi-Levi and D. Williamson, A note on the prize collecting traveling salesman problem, Mathematical Programming, 59 (1993), 413-420. doi: 10.1007/BF01581256.  Google Scholar

[3]

E. Balas and M. Padberg, Set partitioning: a survey, SIAM Review, 18 (1976), 710-760.  Google Scholar

[4]

M. Charikar, S. Khuller, D. Mount and G. Narasimhan, Algorithms for facility location problems with outliers, in Symposium on Discrete Algorithms (SODA), SIAM, (2001), 642-651.  Google Scholar

[5]

D. Du, R. Lu and D. Xu, A primal-dual approximation algorithm for the facility location problem with submodular penalties, Algorithmica, 63 (2012), 191-200. doi: 10.1007/s00453-011-9526-1.  Google Scholar

[6]

D. Du, R. Lu and D. Xu, A primal-dual approximation algorithm for the facility location problem with submodular penalties, Algorithmica, 63 (2012), 191-200. doi: 10.1007/s00453-011-9526-1.  Google Scholar

[7]

J. Edmonds, Submodular functions, matroids, and certain polyhedra, Combinatorial Structures and Their Applications, (1976), 69-87.  Google Scholar

[8]

U. Feige, A threshold of lnn for approximating set-cover, in 28th ACM Symposium on Theory of Computing, (1996), 314-318. doi: 10.1145/237814.237977.  Google Scholar

[9]

L. Fleischer and S. Iwata, A push-relabel framework for submodular function minimization and applications to parametric optimization, Discrete Applied Mathematics, 131 (2003), 311-322. doi: 10.1016/S0166-218X(02)00458-4.  Google Scholar

[10]

S. Fujishige, Submodular Functions and Optimization, 2nd edition, Elsevier, Amsterdam, the North-Holland, 58 (2005).  Google Scholar

[11]

R. Gandhi, S. Khuller and A. Srinivasan, Approximation algorithms for partial covering problems, Journal of Algorithms, 53 (2004), 55-84. doi: 10.1016/j.jalgor.2004.04.002.  Google Scholar

[12]

M. Grötschel, L. Lovász and A. Schrijver, The ellipsoid method and its consequences in combinatorial optimization, Combinatorical, (1981), 169-197. doi: 10.1007/BF02579273.  Google Scholar

[13]

M. Grötschel, L. Lovász and A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, Berlin, 1988. doi: 10.1007/978-3-642-97881-4.  Google Scholar

[14]

R. S. Garfinkel and G. L. Nemhauser, Optimal set covering: a survey, Perspectives on Optimization, (1972), 164-183. Google Scholar

[15]

M. X. Goemans and D. P. Williamson, A general approximation technique for constrained forest problems, SIAM Journal on Computing, 24 (1995), 296-317. doi: 10.1137/S0097539793242618.  Google Scholar

[16]

D. S. Hochbaum, Approximating covering and packing problems: set cover, vertex cover,independent set, and related problems, in Approximation Algorithms for NP-Hard Problems, chapter 3 (eds. D. S. Hochbaum), PWS Publishing Company, (1997), 94-143. Google Scholar

[17]

D. S. Hochbaum, Approximation algorithms for the set covering and vertex cover problems, SIAM Journal on Computing, 11 (1982), 555-556. doi: 10.1137/0211045.  Google Scholar

[18]

S. Iwata, A faster scaling algorithm for minimizing submodular functions, SIAM Journal on Computing, 32 (2003), 833-840. doi: 10.1137/S0097539701397813.  Google Scholar

[19]

S. Iwata, L. Fleischer and S. Fujishige, A combinatorial strongly polynomial algorithm for minimizing submodular functions, Journal of the ACM, 48 (2001), 761-777. doi: 10.1145/502090.502096.  Google Scholar

[20]

S. Iwata and K. Nagano, Submodular function minimization under covering constraints, in the 50th Annual IEEE Symposium on Foundations of Computer Science, IEEE Press, Atlanta, (2009), 671-680. doi: 10.1109/FOCS.2009.31.  Google Scholar

[21]

S. Iwata and J. B. Orlin, A simple combinatorial algorithm for submodular function minimization, in the 20th Annual ACM-SIAM Symposium on Discrete Algorithms, (2009), 1230-1237.  Google Scholar

[22]

R. M. Karp, Reducibility among combinatorial problems, in Complexity of Computer Computations (eds. R.E. Miller and J.W. Thatcher), Plenum Press, (1972), 85-103.  Google Scholar

[23]

A. Krause and C. Guestrin, Near-optimal observation selection using submodular functions, in the 22nd national conference on Artificial intelligence, AAAI, (2007), 1650-1654. Google Scholar

[24]

P. Kohli, P. Kumar and P. Torr, P3 beyond: move making algorithms for solving higher order functions, IEEE Transactions on Pattern Analysis and Machine Intelligence, 31 (2009), 1645-1656. Google Scholar

[25]

J. Könemann, O. Parekh and D. Segev, A unified approach to approximating partial covering problems, Algorithmica, 59 (2011), 489-509. doi: 10.1007/s00453-009-9317-0.  Google Scholar

[26]

H. Lin and J. Bilmes, A class of submodular functions for document summarization, In: In North American chapter of the Association for Computational Linguistics/Human Language Technology Conference, (NAACL/HLT-2011), (2011). Google Scholar

[27]

Y. Li, D. Du, N. Xiu and D. Xu, Improved approximation algorithms for the facility location problems with linear/submodular penalty, in the 19th International Computing and Combinatorics Conference, 7936 (2013), 292-303. doi: 10.1007/978-3-642-38768-5_27.  Google Scholar

[28]

L. Lovász, Submodular functions and convexity, in Mathematical Programming the State of the Art (eds. A. Bachem, M. Grtschel and B. Korte), Springer, (1983), 235-257.  Google Scholar

[29]

J. B. Orlin, A faster strongly polynomial time algorithm for submodular function minimization, Mathematical Programming, 118 (2009), 237-251. doi: 10.1007/s10107-007-0189-2.  Google Scholar

[30]

M. W. Padberg, Covering, packing and knapsack problems, Annals of Discrete Mathematics, 4 (1979), 265-287.  Google Scholar

[31]

A. Schrijver, A combinatorial algorithm minimizing submodular functions in strongly polynomial time, Journal of Combinatorial Theory, Series B, 80 (2000), 346-355. doi: 10.1006/jctb.2000.1989.  Google Scholar

[32]

A. Schrijver, Combinatorial Optimization : Polyhedra and Efficiency, Volume B, Part IV, Chapters 39-49, Springer, 2003.  Google Scholar

[33]

R. Raz and S. Safra, A sub-constant error-probability low-degree test, and a sub-constant error-probability pep characterization of np, in the 29th Annual ACM Symposium on Theory of Computing, (1997), 475-484.  Google Scholar

[34]

D. P. Williamson and D. B Shmoys, The Design of Approximation Algorithms, Cambridge University Press, New York, 2011. Google Scholar

[35]

D. Xu, F. Wang, D. Du and C. Wu, Primal-dual approximation algorithms for submodular vertex cover problems with linear/submodular penalties, in the 19th International Computing and Combinatorics Conference, 8591 (2014), 336-345. Google Scholar

show all references

References:
[1]

E. Balas, The prize collecting traveling salesman problem, Networks, 19 (1989), 621-636. doi: 10.1002/net.3230190602.  Google Scholar

[2]

D. Bienstock, M. Goemans, D. Simchi-Levi and D. Williamson, A note on the prize collecting traveling salesman problem, Mathematical Programming, 59 (1993), 413-420. doi: 10.1007/BF01581256.  Google Scholar

[3]

E. Balas and M. Padberg, Set partitioning: a survey, SIAM Review, 18 (1976), 710-760.  Google Scholar

[4]

M. Charikar, S. Khuller, D. Mount and G. Narasimhan, Algorithms for facility location problems with outliers, in Symposium on Discrete Algorithms (SODA), SIAM, (2001), 642-651.  Google Scholar

[5]

D. Du, R. Lu and D. Xu, A primal-dual approximation algorithm for the facility location problem with submodular penalties, Algorithmica, 63 (2012), 191-200. doi: 10.1007/s00453-011-9526-1.  Google Scholar

[6]

D. Du, R. Lu and D. Xu, A primal-dual approximation algorithm for the facility location problem with submodular penalties, Algorithmica, 63 (2012), 191-200. doi: 10.1007/s00453-011-9526-1.  Google Scholar

[7]

J. Edmonds, Submodular functions, matroids, and certain polyhedra, Combinatorial Structures and Their Applications, (1976), 69-87.  Google Scholar

[8]

U. Feige, A threshold of lnn for approximating set-cover, in 28th ACM Symposium on Theory of Computing, (1996), 314-318. doi: 10.1145/237814.237977.  Google Scholar

[9]

L. Fleischer and S. Iwata, A push-relabel framework for submodular function minimization and applications to parametric optimization, Discrete Applied Mathematics, 131 (2003), 311-322. doi: 10.1016/S0166-218X(02)00458-4.  Google Scholar

[10]

S. Fujishige, Submodular Functions and Optimization, 2nd edition, Elsevier, Amsterdam, the North-Holland, 58 (2005).  Google Scholar

[11]

R. Gandhi, S. Khuller and A. Srinivasan, Approximation algorithms for partial covering problems, Journal of Algorithms, 53 (2004), 55-84. doi: 10.1016/j.jalgor.2004.04.002.  Google Scholar

[12]

M. Grötschel, L. Lovász and A. Schrijver, The ellipsoid method and its consequences in combinatorial optimization, Combinatorical, (1981), 169-197. doi: 10.1007/BF02579273.  Google Scholar

[13]

M. Grötschel, L. Lovász and A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, Berlin, 1988. doi: 10.1007/978-3-642-97881-4.  Google Scholar

[14]

R. S. Garfinkel and G. L. Nemhauser, Optimal set covering: a survey, Perspectives on Optimization, (1972), 164-183. Google Scholar

[15]

M. X. Goemans and D. P. Williamson, A general approximation technique for constrained forest problems, SIAM Journal on Computing, 24 (1995), 296-317. doi: 10.1137/S0097539793242618.  Google Scholar

[16]

D. S. Hochbaum, Approximating covering and packing problems: set cover, vertex cover,independent set, and related problems, in Approximation Algorithms for NP-Hard Problems, chapter 3 (eds. D. S. Hochbaum), PWS Publishing Company, (1997), 94-143. Google Scholar

[17]

D. S. Hochbaum, Approximation algorithms for the set covering and vertex cover problems, SIAM Journal on Computing, 11 (1982), 555-556. doi: 10.1137/0211045.  Google Scholar

[18]

S. Iwata, A faster scaling algorithm for minimizing submodular functions, SIAM Journal on Computing, 32 (2003), 833-840. doi: 10.1137/S0097539701397813.  Google Scholar

[19]

S. Iwata, L. Fleischer and S. Fujishige, A combinatorial strongly polynomial algorithm for minimizing submodular functions, Journal of the ACM, 48 (2001), 761-777. doi: 10.1145/502090.502096.  Google Scholar

[20]

S. Iwata and K. Nagano, Submodular function minimization under covering constraints, in the 50th Annual IEEE Symposium on Foundations of Computer Science, IEEE Press, Atlanta, (2009), 671-680. doi: 10.1109/FOCS.2009.31.  Google Scholar

[21]

S. Iwata and J. B. Orlin, A simple combinatorial algorithm for submodular function minimization, in the 20th Annual ACM-SIAM Symposium on Discrete Algorithms, (2009), 1230-1237.  Google Scholar

[22]

R. M. Karp, Reducibility among combinatorial problems, in Complexity of Computer Computations (eds. R.E. Miller and J.W. Thatcher), Plenum Press, (1972), 85-103.  Google Scholar

[23]

A. Krause and C. Guestrin, Near-optimal observation selection using submodular functions, in the 22nd national conference on Artificial intelligence, AAAI, (2007), 1650-1654. Google Scholar

[24]

P. Kohli, P. Kumar and P. Torr, P3 beyond: move making algorithms for solving higher order functions, IEEE Transactions on Pattern Analysis and Machine Intelligence, 31 (2009), 1645-1656. Google Scholar

[25]

J. Könemann, O. Parekh and D. Segev, A unified approach to approximating partial covering problems, Algorithmica, 59 (2011), 489-509. doi: 10.1007/s00453-009-9317-0.  Google Scholar

[26]

H. Lin and J. Bilmes, A class of submodular functions for document summarization, In: In North American chapter of the Association for Computational Linguistics/Human Language Technology Conference, (NAACL/HLT-2011), (2011). Google Scholar

[27]

Y. Li, D. Du, N. Xiu and D. Xu, Improved approximation algorithms for the facility location problems with linear/submodular penalty, in the 19th International Computing and Combinatorics Conference, 7936 (2013), 292-303. doi: 10.1007/978-3-642-38768-5_27.  Google Scholar

[28]

L. Lovász, Submodular functions and convexity, in Mathematical Programming the State of the Art (eds. A. Bachem, M. Grtschel and B. Korte), Springer, (1983), 235-257.  Google Scholar

[29]

J. B. Orlin, A faster strongly polynomial time algorithm for submodular function minimization, Mathematical Programming, 118 (2009), 237-251. doi: 10.1007/s10107-007-0189-2.  Google Scholar

[30]

M. W. Padberg, Covering, packing and knapsack problems, Annals of Discrete Mathematics, 4 (1979), 265-287.  Google Scholar

[31]

A. Schrijver, A combinatorial algorithm minimizing submodular functions in strongly polynomial time, Journal of Combinatorial Theory, Series B, 80 (2000), 346-355. doi: 10.1006/jctb.2000.1989.  Google Scholar

[32]

A. Schrijver, Combinatorial Optimization : Polyhedra and Efficiency, Volume B, Part IV, Chapters 39-49, Springer, 2003.  Google Scholar

[33]

R. Raz and S. Safra, A sub-constant error-probability low-degree test, and a sub-constant error-probability pep characterization of np, in the 29th Annual ACM Symposium on Theory of Computing, (1997), 475-484.  Google Scholar

[34]

D. P. Williamson and D. B Shmoys, The Design of Approximation Algorithms, Cambridge University Press, New York, 2011. Google Scholar

[35]

D. Xu, F. Wang, D. Du and C. Wu, Primal-dual approximation algorithms for submodular vertex cover problems with linear/submodular penalties, in the 19th International Computing and Combinatorics Conference, 8591 (2014), 336-345. Google Scholar

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