Abstract
This paper addresses the problem of generating synthetic test cases for experimentation in linear programming. We propose a method which maps instance generation and instance space search to an alternative encoded space. This allows us to develop a generator for feasible bounded linear programming instances with controllable properties. We show that this method is capable of generating any feasible bounded linear program, and that parameterised generators and search algorithms using this approach generate only feasible bounded instances. Our results demonstrate that controlled generation and instance space search using this method achieves feature diversity more effectively than using a direct representation.








Similar content being viewed by others
References
Asahiro, Y., Iwama, K., Miyano, E.: Random generation of test instances with controlled attributes. DIMACS Ser. Discrete Math. Theor. Comput. Sci. 26, 377–393 (1996)
Bischl, B., Kerschke, P., Kotthoff, L., Lindauer, M., Malitsky, Y., Fréchette, A., Hoos, H., Hutter, F., Leyton-Brown, K., Tierney, K., Vanschoren, J.: ASlib: a benchmark library for algorithm selection. Artif. Intell. 237, 41–58 (2016)
Bixby, R.E.: A brief history of linear and mixed-integer programming computation. Documenta Mathematica—Extra Volume ISMP pp. 107–121 (2012)
Bowly, S.: simonbowly/lp-generators: v0.2-beta (Version v0.2-beta). Zenodo. http://dx.doi.org/10.5281/zenodo.1220448 (2018)
Chakraborty, S., Choudhury, P.P.: A statistical analysis of an algorithm’s complexity. Appl. Math. Lett. 13(5), 121–126 (2000)
Cotta, C., Moscato, P.: A mixed evolutionary-statistical analysis of an algorithm’s complexity. Appl. Math. Lett. 16(1), 41–47 (2003)
Culberson, J.: Graph Coloring Page. http://webdocs.cs.ualberta.ca/~joe/Coloring/ (2010). Accessed 03 April 2017
Drugan, M.M.: Instance generator for the quadratic assignment problem with additively decomposable cost function. In: 2013 IEEE Congress on Evolutionary Computation, pp. 2086–2093. IEEE (2013)
Gao, W., Nallaperuma, S., Neumann, F.: Feature-based diversity optimization for problem instance classification. In: International Conference on Parallel Problem Solving from Nature, pp. 869–879. Springer (2016)
Gleixner, A., Hendel, G., Gamrath, G., Achterberg, T., Bastubbe, M., Berthold, T., Christophel, P.M., Jarck, K., Koch, T., Linderoth, J., Lübbecke, M., Mittelmann, H.D., Ozyurt, D., Ralphs, T.K., Salvagnin, D., Shinano, Y.: MIPLIB 2017. https://miplib.zib.de/ (2018). Accessed 30 July 2019
Hall, N.G., Posner, M.E.: The generation of experimental data for computational testing in optimization. In: Experimental Methods for the Analysis of Opimization Algorithms, pp. 73–101. Springer, Berlin Heidelberg (2010)
Hill, R., Moore, J., Hiremath, C., Cho, Y.: Test problem generation of binary knapsack problem variants and the implications of their use. Int. J. Oper. Quant. Manag. 18(2), 105–128 (2011)
Hill, R.R., Reilly, C.H.: The effects of coefficient correlation structure in two-dimensional knapsack problems on solution procedure performance. Manage. Sci. 46(2), 302–317 (2000)
Hooker, J.N.: Needed: an empirical science of algorithms. Oper. Res. 42(2), 201–212 (1994)
Hooker, J.N.: Testing heuristics: we have it all wrong. J. Heuristics 1(1), 33–42 (1995)
Hutter, F., Hoos, H.H., Leyton-Brown, K., Stützle, T.: ParamILS: an automatic algorithm configuration framework. J. Artif. Intell. Res. 36(1), 267–306 (2009)
Hutter, F., Xu, L., Hoos, H.H., Leyton-Brown, K.: Algorithm runtime prediction: methods & evaluation. Artif. Intell. 206(1), 79–111 (2014)
Kadioglu, S., Malitsky, Y., Sellmann, M., Tierney, K.: ISAC-instance-specific algorithm configuration. ECAI 215, 751–756 (2010)
Klingman, D., Napier, A., Stutz, J.: NETGEN: a program for generating large scale capacitated assignment, transportation, and minimum cost flow network problems. Manage. Sci. 20(5), 814–821 (1974)
Koch, T., Achterberg, T., Andersen, E., Bastert, O., Berthold, T., Bixby, R.E., Danna, E., Gamrath, G., Gleixner, A.M., Heinz, S., Lodi, A., Mittelmann, H., Ralphs, T., Salvagnin, D., Steffy, D.E., Wolter, K.: MIPLIB 2010: mixed integer programming library version 5. Math. Program. Comput. 3(2), 103–163 (2011)
Leyton-Brown, K., Nudelman, E., Shoham, Y.: Empirical hardness models: methodology and a case study on combinatorial auctions. J. ACM 56(4), 1–52 (2009)
Lin, S., Kernighan, B.W.: An effective heuristic algorithm for the traveling-salesman problem. Oper. Res. 21(2), 498–516 (1973)
McGeoch, C.C.: Feature article—toward an experimental method for algorithm simulation. INFORMS J. Comput. 8(1), 1–15 (1996)
Pilcher, M.G., Rardin, R.L.: Partial polyhedral description and generation of discrete optimization problems with known optima. Nav. Res. Logist. (NRL) 39(6), 839–858 (1992)
Rice, J.R.: The algorithm selection problem. Adv. Comput. 15, 65–118 (1976)
Smith-Miles, K., Bowly, S.: Generating new test instances by evolving in instance space. Comput. Oper. Res. 63, 102–113 (2015)
Smith-Miles, K., Lopes, L.: Measuring instance difficulty for combinatorial optimization problems. Comput. Oper. Res. 39, 875–889 (2012)
Todd, M.J.: Probabilistic models for linear programming. Math. Oper. Res. 16(4), 671–693 (1991)
Van Hemert, J.I.: Evolving combinatorial problem instances that are difficult to solve. Evol. Comput. 14(4), 433–462 (2006)
Xu, L., Hutter, F., Hoos, H.H., Leyton-Brown, K.: Hydra-MIP: automated algorithm configuration and selection for mixed integer programming. In: RCRA Workshop on Experimental Evaluation of Algorithms for Solving Problems with Combinatorial Explosion at the International Joint Conference on Artificial Intelligence (IJCAI), pp. 16–30 (2011)
Acknowledgements
The authors would like to thank the anonymous referees from Mathematical Programming Computation whose thorough comments significantly improved the focus and quality of this work.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This research is funded by the Australian Research Council under Australian Laureate Fellowship FL140100012.
Rights and permissions
About this article
Cite this article
Bowly, S., Smith-Miles, K., Baatar, D. et al. Generation techniques for linear programming instances with controllable properties. Math. Prog. Comp. 12, 389–415 (2020). https://doi.org/10.1007/s12532-019-00170-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12532-019-00170-6