Aggregation-based cutting-planes for packing and covering integer programs

  • Merve Bodur
  • Alberto Del Pia
  • Santanu S. Dey
  • Marco Molinaro
  • Sebastian Pokutta
Full Length Paper Series A
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Abstract

In this paper, we study the strength of Chvátal–Gomory (CG) cuts and more generally aggregation cuts for packing and covering integer programs (IPs). Aggregation cuts are obtained as follows: given an IP formulation, we first generate a single implied inequality using aggregation of the original constraints, then obtain the integer hull of the set defined by this single inequality with variable bounds, and finally use the inequalities describing the integer hull as cutting-planes. Our first main result is to show that for packing and covering IPs, the CG and aggregation closures can be 2-approximated by simply generating the respective closures for each of the original formulation constraints, without using any aggregations. On the other hand, we use computational experiments to show that aggregation cuts can be arbitrarily stronger than cuts from individual constraints for general IPs. The proof of the above stated results for the case of covering IPs with bounds require the development of some new structural results, which may be of independent interest. Finally, we examine the strength of cuts based on k different aggregation inequalities simultaneously, the so-called multi-row cuts, and show that every packing or covering IP with a large integrality gap also has a large k -aggregation closure rank. In particular, this rank is always at least of the order of the logarithm of the integrality gap.

Keywords

Integer programming Cutting planes Packing Covering Aggregation 

Mathematics Subject Classification

90C10 

Notes

Acknowledgements

Santanu S. Dey would like to acknowledge the support of the NSF Grant CMMI#1149400 and Sebastian Pokutta would like to acknowledge the support of the NSF CAREER Award CMMI-1452463. Marco Molinaro would like to acknowledge the support of the grant CNPq Universal #431480/2016-8.

References

  1. 1.
    Bienstock, D., Zuckerberg, M.: Approximate fixed-rank closures of covering problems. Math. Program. 105(1), 9–27 (2006)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bixby, R.E., Fenelon, M., Gu, Z., Rothberg, E., Wunderling, R.: Mixed-integer programming: a progress report. In: Grötschel, M. (ed.) The Sharpest Cut: The Impact of Manfred Padberg and His Work, chap. 18, pp. 309–326. SIAM, Philadelphia (2004)Google Scholar
  3. 3.
    Carr, R.D., Fleischer, L., Leung, V.J., Phillips, C.A.: Strengthening integrality gaps for capacitated network design and covering problems. In: SODA, pp. 106–115 (2000)Google Scholar
  4. 4.
    Chvátal, V., Cook, W., Hartmann, M.: On cutting-plane proofs in combinatorial optimization. Linear Algebra Appl. 114/115, 455–499 (1989)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Conforti, M., Cornuéjols, G., Zambelli, G.: Integer Programming. Springer, Berlin (2014)CrossRefMATHGoogle Scholar
  6. 6.
    Conforti, M., Del Pia, A., Di Summa, M., Faenza, Y., Grappe, R.: Reverse Chvátal–Gomory rank. SIAM J. Discrete Math. 29(1), 166–181 (2015)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cornuéjols, G., Dawande, M.: A class of hard small 0–1 programs. INFORMS J. Comput. 11, 205–210 (1999)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Dey, S.S., Molinaro, M., Wang, Q.: Analysis of sparse cutting-planes for sparse MILPs with applications to stochastic MILPs. arXiv:1601.00198 (2016)
  9. 9.
    Dey, S.S., Morán R., D.A.: Some properties of convex hulls of integer points contained in general convex sets. Math. Program. 141(1-2), 507–526 (2013)Google Scholar
  10. 10.
    Fischetti, M., Lodi, A.: Optimizing over the first Chvátal closure. Math. Program. 110, 3–20 (2007)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Fukasawa, R., Goycoolea, M.: On the exact separation of mixed integer knapsack cuts. Math. Program. 128(1–2), 19–41 (2011). doi: 10.1007/s10107-009-0284-7 MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Gale, D., Rockwell, R.: The Malinvaud eigenvalue lemma: correction and amplification. Econometrica (pre-1986) 44(6), 1323 (1976)CrossRefMATHGoogle Scholar
  13. 13.
    Goberna, M.A., López, M.A.: Linear Semi-infinite Optimization. Wiley, London (1998)MATHGoogle Scholar
  14. 14.
    Goemans, M.X.: Worst-case comparison of valid inequalities for the TSP. Math. Program. 69, 335–349 (1995)MathSciNetMATHGoogle Scholar
  15. 15.
    Hartmann, M.: Cutting planes and the complexity of the integer hull. Tech. rep., Cornell University Operations Research and Industrial Engineering, Cornell University, Ithaca, NY (1998)Google Scholar
  16. 16.
    Jünger, M., Liebling, T.M., Naddef, D., Nemhauser, G.L., Pulleyblank, W.R., Reinelt, G., Rinaldi, G., Wolsey, L.A. (eds.): 50 Years of Integer Programming 1958–2008—From the Early Years to the State-of-the-Art. Springer, Berlin (2010)Google Scholar
  17. 17.
    Lodi, A.: Mixed integer programming computation. In: Jünger et al. [16], pp. 619–645Google Scholar
  18. 18.
    Marchand, H., Martin, A., Weismantel, R., Wolsey, L.A.: Cutting planes in integer and mixed integer programming. Discrete Appl. Math. 123, 397–446 (2002)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Molinaro, M.: Understanding the strength of general-purpose cutting planes. Ph.D. thesis, Carnegie Mellon University (2013)Google Scholar
  20. 20.
    Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley-Interscience, New York (1988)CrossRefMATHGoogle Scholar
  21. 21.
    Pokutta, S., Schulz, A.S.: On the rank of cutting-plane proof systems. In: Eisenbrand, F, Shepherd, F.B. (eds.) Integer Programming and Combinatorial Optimization, pp. 450–463. Springer, Berlin (2010)Google Scholar
  22. 22.
    Pokutta, S., Stauffer, G.: Lower bounds for the Chvátal–Gomory rank in the 0/1 cube. Oper. Res. Lett. 39(3), 200–203 (2011). doi: 10.1016/j.orl.2011.03.001 MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Richard, J.P.P., Dey, S.S.: The group-theoretic approach in mixed integer programming. In: Jünger et al. [16], chap. 19, pp. 727–801Google Scholar
  24. 24.
    Rockafeller, R.T.: Convex Analysis. Princeton University Press, New Jersey (1970)CrossRefGoogle Scholar
  25. 25.
    Rothvoß, T., Sanità, L.: 0/1 polytopes with quadratic Chvátal rank. In: Integer Programming and Combinatorial Optimization—16th International Conference, IPCO 2013, Valparaíso, Chile, March 18–20, 2013. Proceedings, pp. 349–361 (2013). doi: 10.1007/978-3-642-36694-9_30
  26. 26.
    Schrijver, A.: On cutting planes. Combinatorics 79, 291–296 (1980)MathSciNetMATHGoogle Scholar
  27. 27.
    Singh, M., Talwar, K.: Improving integrality gaps via Chvátal–Gomory rounding. In: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, 13th International Workshop, APPROX 2010, and 14th International Workshop, RANDOM 2010, Barcelona, Spain, September 1–3, 2010. Proceedings, pp. 366–379 (2010). doi: 10.1007/978-3-642-15369-3_28
  28. 28.
    Srinivasan, A.: Improved approximation guarantees for packing and covering integer programs. SIAM J. Comput. 29(2), 648–670 (1999)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Vazirani, V.V.: Approximation Algorithms. Springer, Berlin (2013)Google Scholar
  30. 30.
    Weismantel, R.: On the 0/1 knapsack polytope. Math. Program. 77(3), 49–68 (1997)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Wolsey, L.A.: Faces for a linear inequality in 0–1 variables. Math. Program. 8, 165–178 (1975)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Zemel, E.: Lifting the facets of zero–one polytopes. Math. Program. 15, 268–277 (1978)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany and Mathematical Optimization Society 2017

Authors and Affiliations

  1. 1.Department of Mechanical and Industrial EngineeringUniversity of TorontoTorontoCanada
  2. 2.Department of Industrial and Systems Engineering, Wisconsin Institute for DiscoveryUniversity of Wisconsin-MadisonMadisonUSA
  3. 3.School of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlantaUSA
  4. 4.Computer Science DepartmentPUC-RioRio de JaneiroBrazil

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