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A global optimization algorithm for solving the minimum multiple ratio spanning tree problem

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Abstract

This paper studies the sum-of-ratios version of the classical minimum spanning tree problem. We describe a branch-and-bound algorithm for solving the general version of the problem based on its image space representation. The suggested approach specifically addresses the difficulties arising in the case when the number of ratios exceeds two. The efficacy of our approach is demonstrated on randomly generated complete and sparse graph instances.

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References

  1. Aggarwal V., Aneja Y.P., Nair K.P.K.: Minimal spanning tree subject to a side constraint. Comput. Oper. Res. 9, 287–296 (1982)

    Article  Google Scholar 

  2. Ahuja R.K., Magnanti T.L., Orlin J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice-Hall, Englewood Cliffs, NJ (1993)

    Google Scholar 

  3. Birge J.R., Louveaux F.: Introduction to Stochastic Programming. Springer, Berlin (1997)

    Google Scholar 

  4. Bitran G.R., Magnanti T.L.: Duality and sensitivity analysis for fractional programs. Oper. Res. 24, 675–699 (1976)

    Article  Google Scholar 

  5. Boros E., Hammer P.: Pseudo-boolean optimization. Discret. Appl. Math. 123, 155–225 (2002)

    Article  Google Scholar 

  6. Busygin S., Prokopyev O.A., Pardalos P.M.: Feature selection for consistent biclustering via fractional 0–1 programming. J. Comb. Optim. 10, 7–21 (2005)

    Article  Google Scholar 

  7. Chandrasekaran R.: Ratio spanning trees. Networks 7, 335–342 (1977)

    Article  Google Scholar 

  8. Chandrasekaran R., Aneja Y.P., Nair K.P.K.: Minimal cost reliability ratio spanning tree. Ann. Discret. Math. 11, 53–60 (1981)

    Google Scholar 

  9. Chandrasekaran R., Tamir A.: Polynomial testing of the query is a b ≥  c d? with application to finding a minimal cost reliability ratio spanning tree. Discret. Appl. Math. 9, 117–123 (1984)

    Article  Google Scholar 

  10. Chen, D.Z., Daescu, O., Dai, Y., Katoh, N., Wu, X., Xu, J.: Optimizing the sum of linear fractional functions and applications. In: Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 707–716. ACM, New York (2000)

  11. Dinklebach W.: On nonlinear fractional programming. Manag. Sci. 13, 492–498 (1967)

    Article  Google Scholar 

  12. Elhedhli S.: Exact solution of a class of nonlinear knapsack problems. Oper. Res. Lett. 33, 615–624 (2005)

    Article  Google Scholar 

  13. Falk, J.E., Palocsay, S.W.: Optimizing the sum of linear fractional functions. In: Recent Advances in Global Optimization, pp. 221–258. Princeton University Press, Princeton (1992)

  14. Falk J.E., Palocsay S.W.: Image space analysis of generalized fractional programs. J. Glob. Optim. 4, 63–88 (1994)

    Article  Google Scholar 

  15. Gol’stein E.G.: Dual problems of convex and fractionally-convex programming in functional spaces. Sov. Math. Dokl. 8, 212–216 (1967)

    Google Scholar 

  16. Handler G.Y., Zhang I.: A dual algorithm for the constrained shortest path problem. Networks 10, 293–310 (1980)

    Article  Google Scholar 

  17. Hansen P., Poggi de Aragão M., Ribeiro C.C.: Hyperbolic 0–1 programming and query optimization in information retrieval. Math. Program. 52, 256–263 (1991)

    Google Scholar 

  18. Jungnickel D.: Graphs, Networks and Algorithms. Springer, Berlin (2007)

    Google Scholar 

  19. Liang Z.-A., Huang H.X., Pardalos P.M.: Efficiency conditions and duality for a class of multiobjective fractional programming problems. J. Glob. Optim. 27(4), 447–471 (2003)

    Article  Google Scholar 

  20. Matsumoto M., Nishimura T.: Mersenne Twister: a 623-dimensionally equidistributed uniform pseudo-random number generator. ACM Trans. Model. Comput. Simul. 8, 3–30 (1998)

    Article  Google Scholar 

  21. Maurer, J.: The boost random number library. http://www.boost.org/doc/libs/1_38_0/libs/random/index.html. Accessed 29 Apr (2009)

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

    Article  Google Scholar 

  23. Nesterov Yu.: Introductory Lectures on Convex Optimization: A Basic Course. Springer, Berlin (2004)

    Google Scholar 

  24. Pardalos P.M.: An algorithm for a class of nonlinear fractional problems using ranking of the vertices. BIT Numer. Math. 26(3), 392–395 (1986)

    Article  Google Scholar 

  25. Pardalos P.M., Phillips A.: Global optimization of fractional programming. J. Glob. Optim. 1(2), 173–182 (1991)

    Article  Google Scholar 

  26. Prokopyev O.A.: Fractional zero-one programming. In: Floudas, C., Pardalos, P.M. (eds) Encyclopedia of Optimization, pp. 1091–1094. Springer, Berlin (2009)

    Chapter  Google Scholar 

  27. Prokopyev O.A., Huang H.-Z., Pardalos P.M.: On complexity of unconstrained hyperbolic 0–1 programming problems. Oper. Res. Lett. 33, 312–318 (2005)

    Article  Google Scholar 

  28. Prokopyev O.A., Meneses C., Oliveira C.A.S., Pardalos P.M.: On multiple-ratio hyperbolic 0–1 programming problems. Pac. J. Optim. 1, 327–345 (2005)

    Google Scholar 

  29. Quesada I., Grossmann I.E.: A global optimization algorithm for linear fractional and bilinear programs. J. Glob. Optim. 6(1), 39–76 (1995)

    Article  Google Scholar 

  30. Radzik T.: Parametric flows, weighted means of cuts, and fractional combinatorial optimization. In: Pardalos, P.M. (eds) Complexity in Numerical Optimization, pp. 351–386. World Scientific, Singapore (1993)

    Chapter  Google Scholar 

  31. Radzik T.: Fractional combinatorial optimization. In: Floudas, C.A., Pardalos, P.M. (eds) Encyclopedia of Optimization, pp. 159–161. Kluwer, Dordrecht (2001)

    Google Scholar 

  32. Schaible S.: Duality in fractional programming: a unified approach. Oper. Res. 24, 452–461 (1976)

    Article  Google Scholar 

  33. Schaible S., Shi J.: Fractional programming: the sum-of-ratios case. Optim. Methods Softw. 18, 219–229 (2003)

    Article  Google Scholar 

  34. Siek, J., Lee, L.-Q.: The boost graph library. http://www.boost.org/doc/libs/1_38_0/libs/graph/doc/index.html. Accessed 29 Apr (2009)

  35. Skiscim C.C., Palocsay S.W.: Minimum spanning trees with sums of ratios. J. Glob. Optim. 19, 103–120 (2001)

    Article  Google Scholar 

  36. Skiscim C.C., Palocsay S.W.: The complexity of minimum ratio spanning tree problems. J. Glob. Optim. 30, 335–346 (2004)

    Article  Google Scholar 

  37. Stancu-Minasian I.M.: A sixth bibliography of fractional programming. Optimization 55, 405–428 (2006)

    Article  Google Scholar 

  38. Tawarmalani M., Ahmed S., Sahinidis N.: Global optimization of 0–1 hyperbolic programs. J. Glob. Optim. 24, 385–416 (2002)

    Article  Google Scholar 

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Authors and Affiliations

  1. Microsoft Corporation Business Division, One Microsoft Way, Redmond, WA, 98052, USA

    Oleksii Ursulenko

  2. Department of Industrial and Systems Engineering, 241 Zachry Engineering Center, Texas A&M University, College Station, TX, 77840-3131, USA

    Sergiy Butenko

  3. Department of Industrial Engineering, University of Pittsburgh, 1048 Benedum Hall, Pittsburgh, PA, 15261, USA

    Oleg A. Prokopyev

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  1. Oleksii Ursulenko
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  2. Sergiy Butenko
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  3. Oleg A. Prokopyev
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Correspondence to Oleg A. Prokopyev.

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Ursulenko, O., Butenko, S. & Prokopyev, O.A. A global optimization algorithm for solving the minimum multiple ratio spanning tree problem. J Glob Optim 56, 1029–1043 (2013). https://doi.org/10.1007/s10898-011-9832-9

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  • DOI: https://doi.org/10.1007/s10898-011-9832-9

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