Journal of Global Optimization

, Volume 69, Issue 1, pp 255–282 | Cite as

Fractional 0–1 programming: applications and algorithms

  • Juan S. Borrero
  • Colin Gillen
  • Oleg A. Prokopyev


We consider a class of nonlinear integer optimization problems commonly known as fractional 0–1 programming problems (also, often referred to as hyperbolic 0–1 programming problems), where the objective is to optimize the sum of ratios of affine functions subject to a set of linear constraints. Such problems arise in diverse applications across different fields, and have been the subject of study in a number of papers during the past few decades. In this survey we overview the literature on fractional 0–1 programs including their applications, related computational complexity issues and solution methods including exact, approximation and heuristic algorithms.


Fractional 0–1 programming Hyperbolic 0–1 programming Nonlinear integer optimization Binary optimization 



The authors would like to thank the anonymous Associate Editor and two reviewers for their constructive and helpful comments. The research of Oleg Prokopyev was in part performed while visiting the National Research University Higher School of Economics (Nizhny Novgorod) and partially supported by Laboratory of Algorithms and Technologies for Network Analysis (LATNA).


  1. 1.
    Adams, W., Henry, S.: Base-2 expansions for linearizing products of functions of discrete variables. Oper. Res. 60(6), 1477–1490 (2012)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Adams, W.P., Forrester, R.J.: A simple recipe for concise mixed 0–1 linearizations. Oper. Res. Lett. 33(1), 55–61 (2005)MATHCrossRefGoogle Scholar
  3. 3.
    Adams, W.P., Forrester, R.J.: Linear forms of nonlinear expressions: new insights on old ideas. Oper. Res. Lett. 35(4), 510–518 (2007)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Adams, W.P., Sherali, H.D.: A tight linearization and an algorithm for zero-one quadratic programming problems. Manag. Sci. 32(10), 1274–1290 (1986)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Adams, W.P., Forrester, R.J., Glover, F.W.: Comparisons and enhancement strategies for linearizing mixed 0–1 quadratic programs. Discrete Optim. 1(2), 99–120 (2004)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Alguliev, R.M., Aliguliyev, R.M., Mehdiyev, C.A.: Sentence selection for generic document summarization using an adaptive differential evolution algorithm. Swarm Evol. Comput. 1(4), 213–222 (2011)CrossRefGoogle Scholar
  7. 7.
    Almogy, Y., Levin, O.: A class of fractional programming problems. Oper. Res. 19(1), 57–67 (1971)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Amaldi, E., Bosio, S., Malucelli, F., Yuan, D.: Solving nonlinear covering problems arising in WLAN design. Oper. Res. 59(1), 173–187 (2011)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Amaldi, E., Bosio, S., Malucelli, F.: Hyperbolic set covering problems with competing ground-set elements. Math. Program. 134(2), 323–348 (2012)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Amiri, A., Rolland, E., Barkhi, R.: Bandwidth packing with queuing delay costs: bounding and heuristic solution procedures. Eur. J. Oper. Res. 112(3), 635–645 (1999)MATHCrossRefGoogle Scholar
  11. 11.
    Anzai, Y.: On integer fractional programming. J. Oper. Res. Soc. Jpn. 17(1), 49–66 (1974)MathSciNetMATHGoogle Scholar
  12. 12.
    Arora, S., Puri, M., Swarup, K.: The set covering problem with linear fractional functional. Indian J. Pure Appl. Math. 8(5), 578–588 (1977)MathSciNetMATHGoogle Scholar
  13. 13.
    Avadhanula, V., Bhandari, J., Goyal, V., Zeevi, A.: On the tightness of an LP relaxation for rational optimization and its applications. Oper. Res. Lett. 44(5), 612–617 (2016)Google Scholar
  14. 14.
    Billionnet, A.: Approximate and exact solution methods for the hyperbolic 0–1 knapsack problem. INFOR 40(2), 97 (2002a)Google Scholar
  15. 15.
    Billionnet, A.: Approximation algorithms for fractional knapsack problems. Oper. Res. Lett. 30(5), 336–342 (2002b)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Bitran, G.R., Magnanti, T.L.: Duality and sensitivity analysis for fractional programs. Oper. Res. 24(4), 675–699 (1976)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Boros, E., Hammer, P.: Pseudo-boolean optimization. Discrete Appl. Math. 123(1), 155–225 (2002)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Borrero, J.S., Gillen, C., Prokopyev, O.A.: A simple technique to improve linearized reformulations of fractional (hyperbolic) 0–1 programming problems. Oper. Res. Lett. 44(4), 479–486 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Bront, J.J.M., Méndez-Díaz, I., Vulcano, G.: A column generation algorithm for choice-based network revenue management. Oper. Res. 57(3), 769–784 (2009)MATHCrossRefGoogle Scholar
  20. 20.
    Busygin, S., Prokopyev, O., Pardalos, P.: Feature selection for consistent biclustering via fractional 0–1 programming. J. Comb. Optim. 10(1), 7–21 (2005)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Chandrasekaran, R.: Minimal ratio spanning trees. Networks 7(4), 335–342 (1977)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Chang, C.T.: On the polynomial mixed 0–1 fractional programming problems. Eur. J. Oper. Res. 131(1), 224–227 (2001)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Chaovalitwongse, W., Pardalos, P.M., Prokopyev, O.A.: A new linearization technique for multi-quadratic 0–1 programming problems. Oper. Res. Lett. 32(6), 517–522 (2004)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Charnes, A., Cooper, W.W.: Programming with linear fractional functionals. Naval Res. Logist. Q. 10(1), 273–274 (1963)CrossRefGoogle Scholar
  25. 25.
    Cooper, M.W.: A survey of methods for pure nonlinear integer programming. Manag. Sci. 27(3), 353–361 (1981)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Correa, J.R., Fernandes, C.G., Wakabayashi, Y.: Approximating a class of combinatorial problems with rational objective function. Math. Program. 124(1–2), 255–269 (2010)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Dantzig, G.B., Blattner, W., Rao, M.: Finding a cycle in a graph with minimum cost to time ratio with application to a ship routing problem. Technical report, DTIC Document (1966)Google Scholar
  28. 28.
    Dasdan, A., Gupta, R.K.: Faster maximum and minimum mean cycle algorithms for system-performance analysis. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 17(10), 889–899 (1998)CrossRefGoogle Scholar
  29. 29.
    Dasdan, A., Irani, S.S., Gupta, R.K. (1999) Efficient algorithms for optimum cycle mean and optimum cost to time ratio problems. In: Proceedings of the 36th Annual ACM/IEEE Design Automation Conference, ACM, pp. 37–42Google Scholar
  30. 30.
    Davis, J.M., Gallego, G., Topaloglu, H.: Assortment optimization under variants of the nested logit model. Oper. Res. 62(2), 250–273 (2014)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Deineko, V.G., Klinz, B., Woeginger, G.J.: Uniqueness in quadratic and hyperbolic 0–1 programming problems. Oper. Res. Lett. 41(6), 633–635 (2013)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Dinkelbach, W.: On nonlinear fractional programming. Manag. Sci. 13(7), 492–498 (1967)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Elhedhli, S.: Exact solution of a class of nonlinear knapsack problems. Oper. Res. Lett. 33(6), 615–624 (2005)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Elomri, A., Ghaffari, A., Jemai, Z., Dallery, Y.: Coalition formation and cost allocation for joint replenishment systems. Prod. Oper. Manag. 21(6), 1015–1027 (2012)CrossRefGoogle Scholar
  35. 35.
    Ervolina, T.R., McCormick, S.T.: Two strongly polynomial cut cancelling algorithms for minimum cost network flow. Discrete Appl. Math. 46(2), 133–165 (1993)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Falk, J., Palocsay, S.: Image space analysis of generalized fractional programs. J. Global Optim. 4(1), 63–88 (1994)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Fang, S.C., Gao, D.Y., Sheu, R.L., Xing, W.: Global optimization for a class of fractional programming problems. J. Global Optim. 45(3), 337–353 (2009)MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Fox, B.: Letter to the editor—finding minimal cost-time ratio circuits. Oper. Res. 17(3), 546–551 (1969)CrossRefGoogle Scholar
  39. 39.
    Frenk, H., Schaible, S.: Fractional programming. In: Floudas, C.A., Pardalos, P.M. (eds.) Encyclopedia of Optimization, pp. 1080–1091. Springer, Berlin (2009)CrossRefGoogle Scholar
  40. 40.
    Gilmore, P., Gomory, R.: A linear programming approach to the cutting stock problem-part ii. Oper. Res. 11(6), 863–888 (1963)MATHCrossRefGoogle Scholar
  41. 41.
    Glover, F.: Improved linear integer programming formulations of nonlinear integer problems. Manag. Sci. 22(4), 455–460 (1975)MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Glover, F., Woolsey, E.: Technical note-converting the 0–1 polynomial programming problem to a 0–1 linear program. Oper. Res. 22(1), 180–182 (1974)MATHCrossRefGoogle Scholar
  43. 43.
    Goldberg, A.V., Tarjan, R.E.: Finding minimum-cost circulations by canceling negative cycles. J. ACM 36(4), 873–886 (1989)MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Goyal, V., Ravi, R.: An FPTAS for minimizing a class of quasi-concave functions over a convex set. Oper. Res. Lett. 41(2), 191–196 (2013)Google Scholar
  45. 45.
    Granot, D., Granot, F.: On solving fractional (0, 1) programs by implicit enumeration. Can. J. Oper. Res. Inf. Process. 14, 241–249 (1976)MathSciNetMATHGoogle Scholar
  46. 46.
    Grunspan, M., Thomas, M.: Hyperbolic integer programming. Naval Res. Logist. Q. 20(2), 341–356 (1973)MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    Gupte, A., Ahmed, S., Cheon, M.S., Dey, S.: Solving mixed integer bilinear problems using MILP formulations. SIAM J. Optim. 23(2), 721–744 (2013)Google Scholar
  48. 48.
    Hammer, P.L., Rudeanu, S.: Boolean Methods in Operations Research and Related Areas. Springer Science & Business Media, New York (1968)MATHCrossRefGoogle Scholar
  49. 49.
    Han, J., Lee, K., Lee, C., Park, S.: Exact algorithms for a bandwidth packing problem with queueing delay guarantees. INFORMS J. Comput. 25(3), 585–596 (2013)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Hansen, P.: Methods of nonlinear 0–1 programming. Ann. Discrete Math. 5, 53–70 (1979)MathSciNetMATHCrossRefGoogle Scholar
  51. 51.
    Hansen, P., de Aragão, M., Ribeiro, C.: Boolean query optimization and the 0–1 hyperbolic sum problem. Ann. Math. Artif. Intell. 1(1–4), 97–109 (1990)MATHCrossRefGoogle Scholar
  52. 52.
    Hansen, P., de Aragão, M., Ribeiro, C.: Hyperbolic 0–1 programming and query optimization in information retrieval. Math. Program. 52(1–3), 255–263 (1991)MathSciNetMATHCrossRefGoogle Scholar
  53. 53.
    Hansen, P., Jaumard, B., Mathon, V.: Constrained nonlinear 0–1 programming. ORSA J. Comput. 5(2), 97–119 (1993)MathSciNetMATHCrossRefGoogle Scholar
  54. 54.
    Hartmann, M., Orlin, J.B.: Finding minimum cost to time ratio cycles with small integral transit times. Networks 23(6), 567–574 (1993)MathSciNetMATHCrossRefGoogle Scholar
  55. 55.
    Hashizume, S., Fukushima, M., Katoh, N., Ibaraki, T.: Approximation algorithms for combinatorial fractional programming problems. Math. Program. 37(3), 255–267 (1987)MathSciNetMATHCrossRefGoogle Scholar
  56. 56.
    Ibaraki, T.: Integer programming formulation of combinatorial optimization problems. Discrete Math. 16(1), 39–52 (1976)MathSciNetMATHCrossRefGoogle Scholar
  57. 57.
    Ibaraki, T.: Parametric approaches to fractional programs. Math. Program. 26(3), 345–362 (1983)MathSciNetMATHCrossRefGoogle Scholar
  58. 58.
    Isbell, J., Marlow, W.: Attrition games. Naval Res. Logist. Q. 3(1–2), 71–94 (1956)MathSciNetMATHCrossRefGoogle Scholar
  59. 59.
    Ishii, H., Ibaraki, T., Mine, H.: Fractional knapsack problems. Math. Program. 13(1), 255–271 (1977)MathSciNetMATHCrossRefGoogle Scholar
  60. 60.
    Ito, K., Parhi, K.K.: Determining the minimum iteration period of an algorithm. J. VLSI Signal Process. Syst. Signal Image Video Technol. 11(3), 229–244 (1995)CrossRefGoogle Scholar
  61. 61.
    Iwano, K., Misono, S., Tezuka, S., Fujishige, S.: A new scaling algorithm for the maximum mean cut problem. Algorithmica 11(3), 243–255 (1994)MathSciNetMATHCrossRefGoogle Scholar
  62. 62.
    Karp, R.M.: A characterization of the minimum cycle mean in a digraph. Discrete Math. 23(3), 309–311 (1978)MathSciNetMATHCrossRefGoogle Scholar
  63. 63.
    Kleinrock, L.: Queueing Systems, Volume I: Theory. Wiley Interscience, New York (1975)MATHGoogle Scholar
  64. 64.
    Kochenberger, G., Hao, J.K., Glover, F., Lewis, M., Lü, Z., Wang, H., Wang, Y.: The unconstrained binary quadratic programming problem: a survey. J. Comb. Optim. 28(1), 58–81 (2014)MathSciNetMATHCrossRefGoogle Scholar
  65. 65.
    Lawler, E.L.: Optimal cycles in graphs and the minimal cost-to-time ratio problem. In: Periodic Optimization, pp. 37–60. Springer, Berlin (1972)Google Scholar
  66. 66.
    Lawler, E.L.: Combinatorial optimization: networks and matroids. Holt, Rinehart and Winston (1976), reprinted by Dover Publications, Mineola, NY (2001)Google Scholar
  67. 67.
    Li, H.L.: A global approach for general 0–1 fractional programming. Eur. J. Oper. Res. 73(3), 590–596 (1994a)MATHCrossRefGoogle Scholar
  68. 68.
    Li, H.L.: Global optimization for mixed 0–1 programs with convex or separable continuous functions. J. Oper. Res. Soc. 45(9), 1068–1076 (1994b)MATHCrossRefGoogle Scholar
  69. 69.
    Little, J.D.: A proof for the queuing formula: \(L= \lambda w\). Oper. Res. 9(3), 383–387 (1961)MathSciNetMATHCrossRefGoogle Scholar
  70. 70.
    Martos, B., Whinston, V., et al.: Hyperbolic programming. Naval Res. Logist. Q. 11(2), 135–155 (1964)MathSciNetMATHCrossRefGoogle Scholar
  71. 71.
    McCormick, S.T., Ervolina, T.R.: Computing maximum mean cuts. Discrete Appl. Math. 52(1), 53–70 (1994)MathSciNetMATHCrossRefGoogle Scholar
  72. 72.
    Megiddo, N.: Combinatorial optimization with rational objective functions. Math. Oper. Res. 4(4), 414–424 (1979)MathSciNetMATHCrossRefGoogle Scholar
  73. 73.
    Megiddo, N.: Applying parallel computation algorithms in the design of serial algorithms. J. ACM 30(4), 852–865 (1983)MathSciNetMATHCrossRefGoogle Scholar
  74. 74.
    Méndez-Díaz, I., Miranda-Bront, J., Vulcano, G., Zabala, P.: A branch-and-cut algorithm for the latent-class logit assortment problem. Discrete Appl. Math. 164, 246–263 (2014)MathSciNetMATHCrossRefGoogle Scholar
  75. 75.
    Moeini, M.: The maximum ratio clique problem: a continuous optimization approach and some new results. In: HoaiAn, L.T., Tao, P.D., Nguyen, N.T. (eds.) Modelling, Computation and Optimization in Information Systems and Management Sciences, pp. 215–227. Springer, Berlin (2015)Google Scholar
  76. 76.
    Nagih, A., Plateau, G.: A partition algorithm for 0–1 unconstrained hyperbolic programs. Investig. Oper. 9(1), 2 (2000)MATHGoogle Scholar
  77. 77.
    Nemhauser, G.L., Wolsey, L.A.: Integer Programming and Combinatorial Optimization. Wiley, Chichester (1988)MATHGoogle Scholar
  78. 78.
    Nemhauser, G.L., Savelsbergh, M.W.P., Sigismondi, G.S.: Constraint classification for mixed integer programming formulations COAL. Bulletin 20, 8–12 (1992)Google Scholar
  79. 79.
    Nouri, M., Ghodsi, M.: Scheduling tasks with exponential duration on unrelated parallel machines. Discrete Appl. Math. 160(16), 2462–2473 (2012)MathSciNetMATHCrossRefGoogle Scholar
  80. 80.
    Nowozin, S.: Optimal decisions from probabilistic models: the intersection-over-union case. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 548–555 (2014)Google Scholar
  81. 81.
    Orlin, J.B., Ahuja, R.K.: New scaling algorithms for the assignment and minimum mean cycle problems. Math. Program. 54(1–3), 41–56 (1992)MathSciNetMATHCrossRefGoogle Scholar
  82. 82.
    Pardalos, P.M., Phillips, A.: Global optimization of fractional programs. J. Global Optim. 1(2), 173–182 (1991)MathSciNetMATHCrossRefGoogle Scholar
  83. 83.
    Picard, J.C., Queyranne, M.: A network flow solution to some nonlinear 0–1 programming problems, with applications to graph theory. Networks 12(2), 141–159 (1982)MathSciNetMATHCrossRefGoogle Scholar
  84. 84.
    Prokopyev, O.: Fractional zero-one programming. In: Floudas, C.A., Pardalos, P.M. (eds.) Encyclopedia of Optimization, pp. 1091–1094. Springer, Berlin (2008)Google Scholar
  85. 85.
    Prokopyev, O., Huang, H.X., Pardalos, P.: On complexity of unconstrained hyperbolic 0–1 programming problems. Oper. Res. Lett. 33(3), 312–318 (2005a)MathSciNetMATHCrossRefGoogle Scholar
  86. 86.
    Prokopyev, O., Meneses, C., Oliveira, C., Pardalos, P.: On multiple-ratio hyperbolic 0–1 programming problems. Pac. J. Optim. 1(2), 327–345 (2005b)MathSciNetMATHGoogle Scholar
  87. 87.
    Prokopyev, O.A., Kong, N., Martinez-Torres, D.L.: The equitable dispersion problem. Eur. J. Oper. Res. 197(1), 59–67 (2009)MathSciNetMATHCrossRefGoogle Scholar
  88. 88.
    Quesada, I., Grossmann, I.E.: A global optimization algorithm for linear fractional and bilinear programs. J. Global Optim. 6(1), 39–76 (1995)MathSciNetMATHCrossRefGoogle Scholar
  89. 89.
    Radzik, T.: Newton’s method for fractional combinatorial optimization. In: Proceedings 33rd Annual Symposium on Foundations of Computer Science, 1992, IEEE, pp. 659–669 (1992)Google Scholar
  90. 90.
    Radzik, T.: Parametric flows, weighted means of cuts, and fractional combinatorial optimization. Complex. Numer. Optim., 351–386 (1993)Google Scholar
  91. 91.
    Radzik, T.: Fractional combinatorial optimization. In: Floudas, C.A., Pardalos, P.M. (eds.) Encyclopedia of Optimization, pp. 1077–1079. Springer, Berlin (2009)CrossRefGoogle Scholar
  92. 92.
    Radzik, T.: Fractional combinatorial optimization. In: Pardalos, P.M., Du, D.Z., Graham, R.L. (eds.) Handbook of Combinatorial Optimization, pp. 1311–1355. Springer, Berlin (2013)CrossRefGoogle Scholar
  93. 93.
    Robillard, P.: (0, 1) hyperbolic programming problems. Naval Res. Logist. Q. 18(1), 47–57 (1971)MathSciNetMATHCrossRefGoogle Scholar
  94. 94.
    Robillard, P., Florian, M.: Hyperbolic Programming with Bivalent Variables. Département d’Informatique, Montreal Université, Publication #41 (1970)Google Scholar
  95. 95.
    Rolland, E., Amiri, A., Barkhi, R.: Queueing delay guarantees in bandwidth packing. Comput. Oper. Res. 26(9), 921–935 (1999)MathSciNetMATHCrossRefGoogle Scholar
  96. 96.
    Saipe, A.: Solving a (0, 1) hyperbolic program by branch and bound. Naval Res. Logist. Q. 22(3), 497–515 (1975)MATHCrossRefGoogle Scholar
  97. 97.
    Schaible, S.: Fractional programming. II, on Dinkelbach’s algorithm. Manag. Sci. 22(8), 868–873 (1976)Google Scholar
  98. 98.
    Schaible, S.: Fractional programming. Zeitschrift für. Oper. Res. 27(1), 39–54 (1983)MathSciNetMATHGoogle Scholar
  99. 99.
    Schaible, S.: Fractional programming. In: Horst, R., Pardalos, P.M. (eds.) Handbook of Global Optimization, pp. 495–608. Springer, Berlin (1995)CrossRefGoogle Scholar
  100. 100.
    Schaible, S., Shi, J.: Fractional programming: the sum-of-ratios case. Optim. Methods Softw. 18(2), 219–229 (2003)MathSciNetMATHCrossRefGoogle Scholar
  101. 101.
    Sethuraman, S., Butenko, S.: The maximum ratio clique problem. Comput. Manag. Sci. 12(1), 197–218 (2015)MathSciNetMATHCrossRefGoogle Scholar
  102. 102.
    Sherali, H., Smith, J.: An improved linearization strategy for zero-one quadratic programming problems. Optim. Lett. 1(1), 33–47 (2007)MathSciNetMATHCrossRefGoogle Scholar
  103. 103.
    Sherali, H.D., Adams, W.P.: A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems, vol. 31. Springer Science & Business Media, Berlin (2013)MATHGoogle Scholar
  104. 104.
    Sherali, H.D., Tuncbilek, C.H.: A global optimization algorithm for polynomial programming problems using a reformulation-linearization technique. J. Global Optim. 2(1), 101–112 (1992)MathSciNetMATHCrossRefGoogle Scholar
  105. 105.
    Shigeno, M., Saruwatari, Y., Matsui, T.: An algorithm for fractional assignment problems. Discrete Appl. Math. 56(2), 333–343 (1995)MathSciNetMATHCrossRefGoogle Scholar
  106. 106.
    Skiscim, C.C., Palocsay, S.W.: Minimum spanning trees with sums of ratios. J. Global Optim. 19(2), 103–120 (2001)MathSciNetMATHCrossRefGoogle Scholar
  107. 107.
    Skiścim, C.C., Palocsay, S.W.: The complexity of minimum ratio spanning tree problems. J. Global Optim. 30(4), 335–346 (2004)MathSciNetMATHCrossRefGoogle Scholar
  108. 108.
    Stancu-Minasian, I.: Fractional Programming: Theory, Methods and Applications, vol. 409. Springer Science & Business Media, Berlin (2012)MATHGoogle Scholar
  109. 109.
    Subramanian, S., Sherali, H.: A fractional programming approach for retail category price optimization. J. Global Optim. 48(2), 263–277 (2010)MathSciNetMATHCrossRefGoogle Scholar
  110. 110.
    Tawarmalani, M., Ahmed, S., Sahinidis, N.: Global optimization of 0–1 hyperbolic programs. J. Global Optim. 24(4), 385–416 (2002)MathSciNetMATHCrossRefGoogle Scholar
  111. 111.
    Trapp, A., Prokopyev, O.A., Busygin, S.: Finding checkerboard patterns via fractional 0–1 programming. J. Comb. Optim. 20(1), 1–26 (2010)MathSciNetMATHCrossRefGoogle Scholar
  112. 112.
    Trapp, A.C., Konrad, R.A.: Finding diverse optima and near-optima to binary integer programs. IIE Trans. 47(11), 1300–1312 (2015)CrossRefGoogle Scholar
  113. 113.
    Ursulenko, O.: Exact methods in fractional combinatorial optimization. Ph.D. thesis, Texas A&M University (2009)Google Scholar
  114. 114.
    Ursulenko, O., Butenko, S., Prokopyev, O.A.: A global optimization algorithm for solving the minimum multiple ratio spanning tree problem. J. Global Optim. 56(3), 1029–1043 (2013)MathSciNetMATHCrossRefGoogle Scholar
  115. 115.
    Vielma, J., Nemhauser, G.: Modeling disjunctive constraints with a logarithmic number of binary variables and constraints. Math. Program. 128(1–2), 49–72 (2011)MathSciNetMATHCrossRefGoogle Scholar
  116. 116.
    Wang, Q., Yang, X., Zhang, J.: A class of inverse dominant problems under weighted l8 norm and an improved complexity bound for Radzik’s algorithm. J. Global Optim. 34(4), 551–567 (2006)MathSciNetMATHCrossRefGoogle Scholar
  117. 117.
    Wang, R.: On the sum-product ratio problem and its applications. Oper. Res. Lett. 44(3), 409–414 (2016)MathSciNetCrossRefGoogle Scholar
  118. 118.
    Watters, L.: Reduction of integer polynomial programming problems to zero-one linear programming problems. Oper. Res. 15(6), 1171–1174 (1967)CrossRefGoogle Scholar
  119. 119.
    Williams, H.: Experiments in the Formulation of Integer Programming Problems. Springer, Berlin (1974)MATHCrossRefGoogle Scholar
  120. 120.
    Wu, T.H.: A note on a global approach for general 0–1 fractional programming. Eur. J. Oper. Res. 101(1), 220–223 (1997)MATHCrossRefGoogle Scholar
  121. 121.
    Young, N.E., Tarjant, R.E., Orlin, J.B.: Faster parametric shortest path and minimum-balance algorithms. Networks 21(2), 205–221 (1991)MathSciNetMATHCrossRefGoogle Scholar
  122. 122.
    Yue, D., Guillén-Gosálbez, G., You, F.: Global optimization of large-scale mixed-integer linear fractional programming problems: A reformulation-linearization method and process scheduling applications. AIChE J. 59(11), 4255–4272 (2013)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Juan S. Borrero
    • 1
  • Colin Gillen
    • 1
  • Oleg A. Prokopyev
    • 1
    • 2
  1. 1.Department of Industrial EngineeringUniversity of PittsburghPittsburghUSA
  2. 2.Laboratory of Algorithms and Technologies for Networks AnalysisNational Research University Higher School of EconomicsNizhny NovgorodRussia

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