Skip to main content
SpringerLink
Log in

On Convex Minimization over Base Polytopes

  • Conference paper
Integer Programming and Combinatorial Optimization (IPCO 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4513))

Abstract

This note considers convex optimization problems over base polytopes of polymatroids. We show that the decomposition algorithm for the separable convex function minimization problems helps us give simple sufficient conditions for the rationality of optimal solutions and that it leads us to some interesting properties, including the equivalence of the lexicographically optimal base problem, introduced by Fujishige, and the submodular utility allocation market problem, introduced by Jain and Vazirani. In addition, we develop an efficient implementation of the decomposition algorithm via parametric submodular function minimization algorithms. Moreover, we show that, in some remarkable cases, non-separable convex optimization problems over base polytopes can be solved in strongly polynomial time.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bixby, R.E., Cunningham, W.H., Topkis, D.M.: The partial order of a polymatroid extreme point. Mathematics of Operations Research 10, 367–378 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  2. Chudak, F.A., Nagano, K.: Efficient solutions to relaxations of combinatorial problems with submodular penalties via the Lovász extension and non-smooth convex optimization. In: Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 79–88 (2007)

    Google Scholar 

  3. Dutta, B.: The egalitarian solution and reduced game properties in convex games. International Journal of Game Theory 19, 153–169 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  4. Dutta, B., Ray, D.: A Concept of egalitarianism under participation constraints. Econometrica 57, 615–635 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  5. Edmonds, J.: Submodular functions, matroids, and certain polyhedra. In: Guy, R., Hanai, H., Sauer, N., Schönheim, J. (eds.) Combinatorial Structures and Their Applications, pp. 69–87. Gordon and Breach, New York (1970)

    Google Scholar 

  6. Fleischer, L., Iwata, S.: A push-relabel framework for submodular function minimization and applications to parametric optimization. Discrete Applied Mathematics 131, 311–322 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Fujishige, S.: Lexicographically optimal base of a polymatroid with respect to a weight vector. Mathematics of Operations Research 5, 186–196 (1980)

    MATH  MathSciNet  Google Scholar 

  8. Fujishige, S.: Submodular systems and related topics. Mathematical Programming Study 22, 113–131 (1984)

    MATH  MathSciNet  Google Scholar 

  9. Fujishige, S.: Submodular Functions and Optimization, 2nd edn. Elsevier, Amsterdam (2005)

    MATH  Google Scholar 

  10. Gallo, G., Grigoriadis, M.D., Tarjan, R.E.: A fast parametric maximum flow algorithm and applications. SIAM Journal on Computing 18, 30–55 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  11. Groenevelt, H.: Two algorithms for maximizing a separable concave function over a polymatroid feasible region. European Journal of Operational Research 54, 227–236 (1991)

    Article  MATH  Google Scholar 

  12. Hayrapetyan, A., Swamy, C., Tardos, É.: Network design for information networks. In: Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 933–942 (2005)

    Google Scholar 

  13. Hochbaum, D.S.: Lower and upper bounds for the allocation problem and other nonlinear optimization problems. Mathematics of Operations Research 19, 390–409 (1994)

    MATH  MathSciNet  Google Scholar 

  14. Iwata, S.: A faster scaling algorithm for minimizing submodular functions. SIAM Journal on Computing 32, 833–840 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Jain, K., Vazirani, V.V.: Eisenberg-Gale markets: algorithms and structural properties. In: Proceedings of the 39th ACM Symposium on Theory of Computing, to appear (2007)

    Google Scholar 

  16. Maruyama, F.: A unified study on problems in information theory via polymatroids. Graduation Thesis (In Japanese), University of Tokyo, Japan (1978)

    Google Scholar 

  17. McCormick, S.T.: Submodular function minimization. In: Aardal, K., Nemhauser, G.L., Weismantel, R. (eds.) Discrete Optimization. Handbooks in Operations Research and Management Science, vol. 12, pp. 321–391. Elsevier, Amsterdam (2005)

    Google Scholar 

  18. Megiddo, N.: Optimal flows in networks with multiple sources and sinks. Mathematical Programming 7, 97–107 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  19. Murota, K.: Note on the universal bases of a pair of polymatroids. Journal of Operations Research Society of Japan 31, 565–573 (1988)

    MATH  MathSciNet  Google Scholar 

  20. Murota, K.: Discrete Convex Analysis. SIAM, Philadelphia (2003)

    MATH  Google Scholar 

  21. Orlin, J.B.: A faster strongly polynomial time algorithm for submodular function minimization. In: Fischetti, M., Williamson, D.P. (eds.) IPCO 2007. LNCS, vol. 4513, pp. 240–251. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  22. Schrijver, A.: A combinatorial algorithm minimizing submodular functions in strongly polynomial time. Journal of Combinatorial Theory (B) 80, 346–355 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  23. Sharma, Y., Swamy, C., Williamson, D.P.: Approximation algorithms for prize-collecting forest problems with submodular penalty functions. In: Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1275–1284 (2007)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Tokyo, Tokyo 113-8656, Japan, and, Kyoto University, Kyoto 606-8502, Japan

    Kiyohito Nagano

Authors
  1. Kiyohito Nagano
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Matteo Fischetti David P. Williamson

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer Berlin Heidelberg

About this paper

Cite this paper

Nagano, K. (2007). On Convex Minimization over Base Polytopes. In: Fischetti, M., Williamson, D.P. (eds) Integer Programming and Combinatorial Optimization. IPCO 2007. Lecture Notes in Computer Science, vol 4513. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72792-7_20

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-72792-7_20

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-72791-0

  • Online ISBN: 978-3-540-72792-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation