Skip to main content
SpringerLink
Log in

The LP-rounding plus greed approach for partial optimization revisited

  • Research Article
  • Published:
Frontiers of Computer Science Aims and scope Submit manuscript

Abstract

There are many optimization problems having the following common property: Given a total task consisting of many subtasks, the problem asks to find a solution to complete only part of these subtasks. Examples include the k-Forest problem and the k-Multicut problem, etc. These problems are called partial optimization problems, which are often NP-hard. In this paper, we systematically study the LP-rounding plus greed approach, a method to design approximation algorithms for partial optimization problems. The approach is simple, powerful and versatile. We show how to use this approach to design approximation algorithms for the k-Forest problem, the k-Multicut problem, the k-Generalized connectivity problem, etc.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Agrawal A, Klein P, Ravi R. When trees collide: an approximation algorithm for the generalized Steiner problem on networks. SIAM Journal on Computing, 1995, 24(3): 440–456

    Article  MathSciNet  MATH  Google Scholar 

  2. Goemans M, Williamson D. A general approximation technique for constrained forest problems. SIAM Journal on Computing, 1995, 24(2): 296–317

    Article  MathSciNet  MATH  Google Scholar 

  3. Garg N, Vazirani V, Yannakakis M. Approximate max-flow min-(multi)cut theorems and their applications. SIAM Journal on Computing, 1996, 25: 235–251

    Article  MathSciNet  MATH  Google Scholar 

  4. Garg N, Vazirani V, Yannakakis M. Primal-dual approximation algorithm for integral flow and multicut in trees. Algorithmica, 1997, 18: 3–20

    Article  MathSciNet  MATH  Google Scholar 

  5. Slavík P. Improved performance of the greedy algorithm for partial cover. Information Processing Letters, 1997, 64: 251–254

    Article  MathSciNet  MATH  Google Scholar 

  6. Gupta A, Hajiaghayi M, Nagarajan V, Ravi R. Dial a ride from k-forest. ACM Transactions on Algorithms, 2010, 6(2): 41

    Article  MathSciNet  MATH  Google Scholar 

  7. Hajiaghayi M, Jain K. The prize-collecting generalized Steiner tree problem via a new approach of primal-dual schema. In: Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms. 2006, 631–640

  8. Segev D, Segev G. Approximate k-Steiner forests via the Lagrangian relaxation technique with internal preprocessing. In: Proceedings of the 14th Annual European Symposium on Algorithms. 2006, 600–611

  9. Zhang P, Xia M. An approximation algorithm to the k-Steiner forest problem. Theoretical Computer Science, 2009, 410(11): 1093–1098

    Article  MathSciNet  MATH  Google Scholar 

  10. Zhang P, Zhu D, Luan J. An approximation algorithm for the generalized k-multicut problem. Discrete Applied Mathematics, 2012, 160(7–8): 1240–1247

    Article  MathSciNet  MATH  Google Scholar 

  11. Golovin D, Nagarajan V, Singh M. Approximating the k-multicut problem. In: Proceedings of the 17th ACM-SIAM Symposium on Discrete Algorithms. 2006, 621–630

  12. Levin A, Segev D. Partial multicuts in trees. In: Proceedings of the 3rd Workshop on Approximation and Online Algorithms. 2005, 320–333

  13. Mestre J. Lagrangian relaxation and partial cover (extended abstract). In: Proceedings of the 25th International Symposium on Theoretical Aspects of Computer Science. 2008, 539–550

  14. Räcke H. Optimal hierarchical decompositions for congestion minimization in networks. In: Proceedings of the 40th Annual ACM Symposium on Theory of Computing. 2008, 255–264

  15. Garg N. Saving an epsilon: a 2-approximation for the k-MST problem in graphs. In: Proceedings of the 37th Annual ACM Symposium on Theory of Computing. 2005, 302–309

  16. Johnson D. Approximation algorithms for combinatorial problems. Journal of Computer and System Sciences, 1974, 9: 256–278

    Article  MathSciNet  MATH  Google Scholar 

  17. Lovász L. On the ratio of optimal inegral and factional covers. Discrete Mathematics, 1975, 13: 383–390

    Article  MathSciNet  MATH  Google Scholar 

  18. Chvátal V. A greedy heuristic for the set covering problem. Mathematics of Operations Research, 1979, 4: 233–235

    Article  MathSciNet  MATH  Google Scholar 

  19. Bar-Yehuda R, Even S. A linear-time approximation algorithm for the weighted vertex cover problem. Journal of Algorithms, 1981, 2: 198–203

    Article  MathSciNet  MATH  Google Scholar 

  20. Hochbaum D. Approximation algorithms for the set covering and vertex cover problems. SIAM Journal on Computing, 1982, 11(3): 555–556

    Article  MathSciNet  MATH  Google Scholar 

  21. Gandhi R, Khuller S, Srinivasan A. Approximation algorithms for partial covering problems. Journal of Algorithms, 2004, 53(1): 55–84

    Article  MathSciNet  MATH  Google Scholar 

  22. Hochbaum D. The t-vertex cover problem: extending he half integrality framework with budget constraints. In: Proceedings of the 1st International Workshop on Approximation Algorithms for Combinatorial Optimization Problems. 1998, 111–122

  23. Bshouty N, Burroughs L. Massaging a linear programming solution to give a 2-approximation for a generalization of the vertex cover problem. In: Proceedings of the 15th Annual Symposium on the Theoretical Aspects of Computer Science. 1998, 298–308

  24. Bar-Yehuda R. Using homogeneous weights for approximating the partial cover problem. In: Proceedings of the 10th ACM-SIAM Symposium on Discrete Algorithms. 1999, 71–75

  25. Garg N. A 3-approximation for the minimum tree spanning k vertices. In: Proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science. 1996, 302–309

  26. Jain K, Vazirani V. Approximation algorithms for metric facility location and k-median problems using the primal-dual schema and Lagrangian relaxation. Journal of the ACM, 2001, 48(2): 274–296

    Article  MathSciNet  MATH  Google Scholar 

  27. Chudak F, Roughgarden T, Williamson D. Approximate k-MSTs and k-Steiner trees via the primal-dual method and Lagrangean relaxation. Mathematical Proramming, 2004, 100(2): 411–421

    Article  MathSciNet  MATH  Google Scholar 

  28. Könemann J, Parekh O, Segev D. A unified approach to approximating partial covering problems. Algorithmica, 2011, 59(4): 489–509

    Article  MathSciNet  MATH  Google Scholar 

  29. Gupta A, Kumar A. Greedy algorithms for Steiner forest. In: Proceedings of the 47th Annual ACM Symposium on Theory of Computing. 2015, 871–878

  30. Groß M, Gupta A, Kumar A, Matuschke J, Schmidt D, Schmidt M, Verschae J. A local-search algorithms for Steiner forest. In: Proceedings of the 9th Conference of Innovations in Theoretical Computer Science. 2018

  31. Bhaskara A, Charikar M, Chlamtac E, Feige U, Vijayaraghavan A. Detecting high log-densities: an O(n1/4) approximation for densest k-subgraph. In: Proceedings of the 42nd Annual ACM Symposium on Theory of Computing. 2010, 201–210

  32. Manurangsi P. Almost-polynomial ratio eth-hardness of approximating densest k-subgraph. In: Proceedings of the 49th Annual ACM Symposium on Theory of Computing. 2017, 954–961

  33. Segev D. Approximating k-generalized connectivity via collapsing HSTs. Journal of Combinatorial Optimization, 2011, 21: 364–382

    Article  MathSciNet  MATH  Google Scholar 

  34. Alon N, Awerbuch B, Azar Y, Buchbinder N, Naor J. A general approach to online network optimization problems. ACM Transactions on Algorithms, 2006, 2(4): 640–660

    Article  MathSciNet  MATH  Google Scholar 

  35. Chekuri C, Even G, Gupta A, Segev D. Set connectivity problems in undirected graphs and the directed Steiner network problem. ACM Transactions on Algorithms, 2011, 7(2): 18

    Article  MathSciNet  MATH  Google Scholar 

  36. Gupta A. Improved results for directed multicut. In: Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms. 2003, 454–455

  37. Agarwal A, Alon N, Charikar M. Improved approximation for directed cut problems. In: Proceedings of the 39th Annual ACM Symposium on Theory of Computing. 2007, 671–680

  38. Grandoni F, Rothvoß T. Network design via core detouring for problems without a core. In: Proceedings of the 37th International Colloquium on Automata, Languages and Programming. 2010, 490–502

  39. Chlebík M, Chlebíková J. The Steiner tree problem on graphs: inapproximability results. Theoretical Computer Science, 2008, 406: 207–214

    Article  MathSciNet  MATH  Google Scholar 

  40. Chuzhoy J, Gupta A, Naor J, Sinha A. On the approximability of some network design problems. ACM Transactions on Algorithms, 2008, 4(2): 23:1–23:17

    Article  MathSciNet  MATH  Google Scholar 

  41. Andrews M. Hardness of buy-at-bulk network design. In: Proceedings of the 45th Symposium on Foundations of Computer Science. 2004, 115–124

Download references

Acknowledgements

The work was supported by the National Natural Science Foundation of China (Grant Nos. 61972228, 61672323, 61672328), and the Natural Science Foundation of Shandong Province (ZR2016AM28, ZR2019MF072).

Author information

Authors and Affiliations

  1. School of Software, Shandong University, Jinan, 250101, China

    Peng Zhang

Authors
  1. Peng Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peng Zhang.

Additional information

Peng Zhang is an associate professor of computer science at School of Software, Shandong University, China. He received his PhD degree in computer science from Institute of Software, Chinese Academy of Sciences, China in 2007. His research interests include approximation algorithms, combinatorial optimization and computational complexity. He has published more than fifty papers mainly in approximation algorithms, in top and mainstream journals such as Information and Computation, Algorithmica, Theory of Computing Systems, Theoretical Computer Science, Discrete Applied Mathematics, etc., and in mainstream conferences such as LATIN, ISAAC, COCOON, etc.

Electronic supplementary material

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, P. The LP-rounding plus greed approach for partial optimization revisited. Front. Comput. Sci. 16, 161402 (2022). https://doi.org/10.1007/s11704-020-0368-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11704-020-0368-3

Keywords

Search

Navigation