Skip to main content
SpringerLink
Log in

Fractional Combinatorial Optimization

  • Chapter
Book cover Handbook of Combinatorial Optimization

Abstract

An instance of a fractional combinatorial optimization problem F consists of a specification of a set \(\chi\subseteq{\left\{{0,1}\right\}^p}\), and two functions f : χ → R and g : χ → R. The task is to

$$F:maximize\frac{{f(x)}}{{g(x)}},forx \in X$$

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 259.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 329.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 329.99
Price excludes VAT (USA)
  • Durable hardcover 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. R. E. Gellman. On a routing problem. Quarterly of Applied Mathematics, 16: 87–90, 1958.

    MathSciNet  Google Scholar 

  2. M. Blum, R. W. Floyd, V. Pratt, R. L. Rivest, and R. E. Tarjan. Time bounds for selection. J. Comp. and Syst. Sci., 7 (4): 448–461, 1973.

    Article  MATH  MathSciNet  Google Scholar 

  3. P. J. Carstensen. The Complexity of Some Problems in Parametric,Linear,and Combinatorial Programming. PhD thesis, Department of Mathematics, University of Michigan, Ann Arbor, Michigan, 1983. Doctoral Thesis.

    Google Scholar 

  4. R. Chandrasekaran. Minimum ratio spanning trees. Networks, 7: 335–342, 1977.

    Article  MATH  MathSciNet  Google Scholar 

  5. E. Cohen and N. Megiddo. Maximizing concave functions in fixed dimension. In P. Pardalos, editor, Complexity in Numerical Optimization, pages 74–87. World Scientific, 1993.

    Chapter  Google Scholar 

  6. E. Cohen and N. Megiddo. Strongly polynomial time and NC algorithms for detecting cycles in dynamic graphs. J. Assoc. Comput. Mach., 40 (4): 791–830, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  7. R. Cole. Parallel merge sort. SIAM J. Comput., 17 (4): 770–785, August 1988.

    Article  MATH  MathSciNet  Google Scholar 

  8. T. H. Cormen, C. E. Leiserson, and R. L. Rivest. Introduction to Algorithms. MIT Press, Cambridge, MA, 1990.

    MATH  Google Scholar 

  9. G. B. Dantzig, W. O. Blattner, and M. R. Rao. Finding a cycle in a graph with minimu cost to time ratio with application to a ship routing problem. In P. Rosentiehl, editor, Theory of Graphs, pages 77–84. Gordon and Breach, New York, 1967.

    Google Scholar 

  10. W. Dinkelbach. On nonlinear fractional programming. Management Science, 13: 492–498, 1967.

    Article  MathSciNet  Google Scholar 

  11. T. R. Ervolina and S. T. McCormick. Two strongly polynomial cut cancelling algorithms for minimum cost network flow. Discrete Applied Math., 46: 133–165, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  12. L. R. Ford, Jr. and D. R. Fulkerson. Flows in Networks. Princeton Univ. Press, Princeton, NJ, 1962.

    MATH  Google Scholar 

  13. B. Fox. Finding minimal cost-time ratio circuits. Oper. Res., 17: 546–551, 1969.

    Article  Google Scholar 

  14. M. L. Fredman and R. E. Tarjan. Fibonacci heaps and their uses in improved network optimization algorithms J. Assoc. Comput. Mach., 34: 596–615, 1987.

    Article  MathSciNet  Google Scholar 

  15. M. Goemans. Personal communication, 1992.

    Google Scholar 

  16. A. V. Goldberg. Scaling algorithms for the shortest paths problem. SIAM J. Comput., 24 (3): 494–504, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  17. A. V. Goldberg and R. E. Tarjan. A new approach to the maximum flow problem. J. Assoc. Comput. Mach., 35: 921–940, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  18. A. V. Goldberg and R. E. Tarjan. Finding minimum-cost circulations by canceling negative cycles. J. Assoc. Comput. Mach., 36: 388–397, 1989.

    Article  MathSciNet  Google Scholar 

  19. M. Goldmann On a subset-sum problem. Personal Communication, 1994.

    Google Scholar 

  20. D. Gusfield. Sensitivity analysis for combinatorila optimizaton. Technical Report UCB/ERL M90/22, Electronics Research Laboratory, University of California, Berkeley, May 1980.

    Google Scholar 

  21. D. Gusfield. Parametric combinatorial computing and a problem of program module distribution. J. Assoc. Comput. Mach.,30(3):551–563, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  22. S. Hashizume, M. Fukushima, N. Katoh, and T. Ibaraki. Approximation algorithms for combinatorial fractional programming problems. Math. Programming, 37: 255–267, 1987.

    Article  MathSciNet  Google Scholar 

  23. T. Ibaraki. Parametric approaches to fractional programs. Math. Programming, 26: 345–362, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  24. H. Ishii, T. Ibaraki, and H. Mine. Fractional knapsack problems. Math. Programming, 13: 255–271, 1976.

    Article  MathSciNet  Google Scholar 

  25. K. Iwano, S. Misono, S. Tezuka, and S. Fujishige. A new scaling algorithm for the maximum mean cut problem. Algorithmica,11(3):243–255, 1994.

    Article  MathSciNet  Google Scholar 

  26. R. M. Karp. A Characterization of the minimum cycle mean in a digraph. Discrete Math., 23: 309–311, 1978.

    MATH  MathSciNet  Google Scholar 

  27. V. King, S. Rao, and R. Tarjan. A faster deterministic maximum flow algorithm. J. Alg., 17 (3): 447–474, 1994.

    Article  MathSciNet  Google Scholar 

  28. E. L. Lawler. Optimal cycles in doubly weighted directed linear graphs. In P. Rosentiehl, editor, Theory of Graphs, pages 209–213. Gordon and Breach, New York, 1967.

    Google Scholar 

  29. E. L. Lawler. Combinatorial Optimization: Networks and Matroids. Holt, Reinhart, and Winston, New York, NY., 1976.

    Google Scholar 

  30. S. T. McCormick. A note on approximate binary search algorithms for mean cuts and cycles. UBC Faculty of Commerce Working Paper 92MSC-021, University of British Columbia, Vancouver, Canada, 1992.

    Google Scholar 

  31. N. Megiddo. Combinatorial optimization with rational objective functions. Math. Oper. Res., 4: 414–424, 1979.

    MATH  MathSciNet  Google Scholar 

  32. N. Megiddo. Applying parallel computation algorithms in the design of serial algorithms J. Assoc. Comput. Mach., 30: 852–865, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  33. C. Haibt Norton, S. A. Plotkin, and É. Tardos. Using separation algorithms in fixed dimension. J. Alg., 13: 79–98, 1992.

    Article  MATH  Google Scholar 

  34. C. H. Papadimitriou and K. Steiglitz. Combinatorial Optimization: Algorithms and Complexity. Prentice-Hall, Englewood Cliffs, NJ, 1982.

    MATH  Google Scholar 

  35. P. M. Pardalos and A. T. Phillips. Global optimization of fractional programs. J. Global Opt., 1: 173–182, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  36. T. Radzik. Newton’s method for fractional combinatorial optimization. In Proc. 33rd IEEE Annual Symposium on Foundations of Computer Science, pages 659–669, 1992.

    Chapter  Google Scholar 

  37. T. Radzik. Newton’s method for fractional combinatorial optimization. Technical Report STAN-CS-92–1406, Department of Computer Science, Stanford University, January 1992.

    Google Scholar 

  38. T. Radzik. Parametric flows, weighted means of cuts, and fractional combinatorial optimization. In P. Pardalos, editor, Complexity in Numerical Optimization, pages 351–386. World Scientific, 1993.

    Chapter  Google Scholar 

  39. T. Radzik and A. Goldberg. Tight bounds on the number of minimum-mean cycle cancellations and related results. Algorithmica, 11: 226–242, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  40. S. Schaible. Fractional Programming 2. On Dinkelbach’s algorithm. Management Sci., 22: 868–873, 1976.

    Article  MATH  MathSciNet  Google Scholar 

  41. S. Schaible. A survey of fractional programming. In S. Schaible and W. T. Ziemba, editors, Generalized Concavity in Optimization and Economics, pages 417–440. Academic Press, New York, 1981.

    Google Scholar 

  42. S. Schaible. Bibliography in fractional programming Zeitschrift für Operations Res., 26 (7), 1982.

    Google Scholar 

  43. S. Schaible. Fractional programming. Zeitschrift für Operations Res., 27: 39–54, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  44. S. Schaible and T. Ibaraki. Fractional programming Europ. J. of Operational Research, 12, 1983.

    Google Scholar 

  45. Y. Shiloach and U. Vishkin. An O(n2 log n) Parallel Max-Flow Algorithm. J. Algorithms, 3: 128–146, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  46. S. Toledo. Maximizing non-linear concave functions in fixed dimension. In P. Pardalos, editor, Complexity in Numerical Optimization, pages 429–447. World Scientific, 1993.

    Google Scholar 

  47. L. G. Valiant. Parallelism in comparison problems. SIAM J. Comput., 4: 348–355, 1975.

    Article  MATH  MathSciNet  Google Scholar 

  48. C. Wallacher. A Generalization of the Minimum-mean Cycle Selection Rule in Cycle Canceling Algorithms. Unpublished manuscript, Institut für Angewandte Mathematik, Technische Universität CaroloWilhelmina, Germany, November 1989.

    Google Scholar 

  49. A. C. Yao. An O (| E| log log | V |) algorithm for finding minimum spanning trees. Information Processing Let., 4: 21–23, 1975.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, King’s College London, London, WC2R 2LS, UK

    Tomasz Radzik

Authors
  1. Tomasz Radzik
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. University of Minnesota, Minneapolis, USA

    Ding-Zhu Du

  2. University of Florida, Gainesville, USA

    Panos M. Pardalos

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Kluwer Academic Publishers

About this chapter

Cite this chapter

Radzik, T. (1998). Fractional Combinatorial Optimization. In: Du, DZ., Pardalos, P.M. (eds) Handbook of Combinatorial Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0303-9_6

Download citation

  • DOI: https://doi.org/10.1007/978-1-4613-0303-9_6

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-7987-4

  • Online ISBN: 978-1-4613-0303-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation