Advertisement

An Exponential Lower Bound on the Length of Some Classes of Branch-and-Cut Proofs

  • Sanjeeb Dash
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2337)

Abstract

Branch-and-cut methods are among the more successful techniques for solving integer programming problems. They can also be used to prove that all solutions of an integer program satisfy a given linear inequality. We examine the complexity of branch-and-cut proofs in the context of 0-1 integer programs. We prove an exponential lower bound on the length of branch-and-cut proofs in the case where branching is on the variables and the cutting planes used are lift-and-project cuts (also called simple disjunctive cuts by some authors), Gomory-Chvátal cuts, and cuts arising from the N0 matrix-cut operator of Lovász and Schrijver. A consequence of the lower-bound result in this paper is that branch-and-cut methods of the type described above have exponential running time in the worst case.

Keywords

Integer Program Proof System Gomory Mixed Integer Monotone Boolean Function Exponential Lower Bound 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ajtai, M.: The complexity of the pigeonhole principle. Combinatorica 14 (1994) 417–433MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Alon, N., Boppana, R.: The monotone circuit complexity of Boolean functions. Combinatorica 7 (1987) 1–22MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Andreev, A.E.: On a method for obtaining lower bounds for the complexity of individual monotone functions. Soviet Mathematics-Doklady 31 (1985) 530–534zbMATHGoogle Scholar
  4. 4.
    Balas, E.: Disjunctive programming. Annals of Discrete Mathematics 5 (1979) 3–51MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Balas, E., Ceria, S., Cornuéjols, G.: A lift-and-project cutting plane algorithm for mixed 0-1 programs. Mathematical Programming 58 (1993) 295–324MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Balas, E., Ceria, S., Cornuéjols, G.: Mixed 0-1 programming by lift-and-project in a branch-and-cut framework. Management Science 42 (1996) 1229–1246zbMATHCrossRefGoogle Scholar
  7. 7.
    Balas, E., Ceria, S., Cornuéjols, G., Natraj, G.: Gomory cuts revisited. Operations Research Letters 19 (1996) 1–9MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Beame, P., Pitassi, T.: Propositional proof complexity: past, present, and future. Bulletin of the European Association for Theoretical Computer Science 65 (1989) 66–89Google Scholar
  9. 9.
    Bonet, M., Pitassi, T., Raz, R.: Lower bounds for cutting planes proofs with small coefficients. The Journal of Symbolic Logic 62 (1997) 708–728.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Boppana, R., Sipser, M.: The complexity of finite functions. In: Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity. MIT Press/Elsevier (1990) 757–804Google Scholar
  11. 11.
    Chvátal, V.: Edmonds polytopes and a hierarchy of combinatorial problems. Discrete Mathematics 4 (1973) 305–337MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Chvátal, V.: Hard knapsack problems. Operations Research 28 (1980) 1402–1411MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Chvátal, V.: Cutting-plane proofs and the stability number of a graph. Report Number 84326-OR. Insitut für Ökonometrie und Operations Research, Universität Bonn, Bonn. (1984)Google Scholar
  14. 14.
    Chvátal, V.: Cutting planes in combinatorics. European Journal of Combinatorics 6 (1985) 217–226MathSciNetzbMATHGoogle Scholar
  15. 15.
    Chvátal, V., Cook, W., Hartmann, M.: On cutting plane proofs in combinatorial optimization. Linear Algebra and its Applications 114/115 (1989) 455–499CrossRefGoogle Scholar
  16. 16.
    Chvátal, V., Szemerédi, E.: Many hard examples for resolution. Journal of the Association for Computing Machinery 35 (1988) 759–768MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Cook, S.A., Haken, A.: An exponential lower bound for the size of monotone real circuits. Journal of Computer and System Sciences 58 (1999) 326–335MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Cook, S.A., Reckhow, R.A.: The relative efficiency of propositional proof systems. Journal of Symbolic Logic 44 (1979) 36–50MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Cook, W.: Cutting-plane proofs in polynomial space. Mathematical Programming 47 (1990) 11–18MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Cook, W., Coullard, C.R., Turán, Gy.: On the complexity of cutting-plane proofs. Discrete Applied Mathematics 18 (1987) 25–38MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Cook, W., Cunningham, W., Pulleyblank, W., Schrijver, A.: Combinatorial Optimization. Wiley, New York (1998)zbMATHGoogle Scholar
  22. 22.
    Cook, W., Dash, S.: On the matrix-cut rank of polyhedra. Mathematics of Operations Research 26 (2001) 19–30MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Cook, W., Hartmann, M.: On the complexity of branch and cut methods for the traveling salesman problem. In: Seymour, P.D., Cook, W. (eds.): Polyhedral Combinatorics, Vol. 1, DIMACS Series in Discrete Mathematics and Theoretical Computer Science (1990) 75–82Google Scholar
  24. 24.
    G. Cornuéjols and Li, Y.: Elementary closures for integer programs. Operations Research Letters 28 (2001) 1–8MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Dash, S.: On the matrix cuts of Lovász and Schrijver and their use in integer programming. Ph.D. Thesis. Rice University, Houston, Texas (2001). Available as: Technical Report TR01-08. Rice University (2001)Google Scholar
  26. 26.
    Eisenbrand, F., Schulz, A.S.: Bounds on the Chvátal rank of polytopes in the 0/1-cube. In: Cornuéjols, G., Burkard, R.E., Woeginger, G.J. (eds.): Integer Programming and Combinatorial Optimization, 7th International IPCO Conference. Springer, Berlin (1999) 137–150CrossRefGoogle Scholar
  27. 27.
    Goemans, M.X., Tunçel, L.: When does the positive semidefiniteness constraint help in lifting procedures. Mathematics of Operations Research 26 (2001) 796–815MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Gomory, R.E.: Outline of an algorithm for integer solutions to linear programs. Bulletin of the American Mathematical Society 64 (1958) 275–278MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Gomory, R.E.: An algorithm for the mixed integer problem. RM-2597, The Rand Corporation (1960)Google Scholar
  30. 30.
    Grötschel, M., Pulleyblank, W.R.: Clique tree inequalities and the symmetric traveling salesman problem. Mathematics of Operations Research 11 (1986) 1–33MathSciNetCrossRefGoogle Scholar
  31. 31.
    Haken, A.: The intractability of resolution. Theoretical Computer Science 39 (1985) 297–308MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press (1985)Google Scholar
  33. 33.
    Jeroslow, R.G.: Trivial integer programs unsolvable by branch-and-bound. Mathematical Programming 6 (1974) 105–109zbMATHCrossRefGoogle Scholar
  34. 34.
    Jeroslow, R.G.: A cutting-plane game for facial disjunctive programs. SIAM Journal on Control and Optimization 18 (1980) 264–281MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Krajíček, J.: Lower bounds to the size of constant-depth propositional proofs. Journal of Symbolic Logic 59 (1994) 73–86MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Krajíček, J.: Interpolation theorems, lower bounds for proof systems and independence results for bounded arithmetic. Journal of Symbolic Logic 62 (1997) 457–486MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Lovász, L.: Stable sets and polynomials. Discrete Mathematics 124 (1994) 137–153MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Lovász, L., Schrijver, A.: Cones of matrices and set-functions and 0-1 optimization. SIAM Journal on Optimization 1 (1991) 166–190MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Pudlák, P.: Lower bounds for resolution and cutting plane proofs and monotone computations. Journal of Symbolic Logic 62 (1997) 981–998MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Pudlák, P.: On the complexity of propositional calculus. In: Sets and Proofs, Invited papers from Logic Colloquium 1997. Cambridge University Press (1999) 197–218Google Scholar
  41. 41.
    Razborov, A.A., Lower bounds for the monotone complexity of some boolean functions. Dokladi Akademii Nauk SSSR 281 (1985) 798–801 (in Russian). English translation in: Soviet Mathematics-Doklady 31 (1985) 354–357MathSciNetGoogle Scholar
  42. 42.
    Razborov, A.A., Unprovability of lower bounds on the circuit size in certain fragments of bounded arithmetic. Izvestiiya of the RAN 59 (1995) 201–224MathSciNetGoogle Scholar
  43. 43.
    Schrijver, A.: On cutting planes. In: Deza, M., Rosenberg, I.G. (eds.): Combinatorics 79 Part II, Annals of Discrete Mathematics 9. North Holland, Amsterdam (1980) 291–296Google Scholar
  44. 44.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley, Chichester (1986)zbMATHGoogle Scholar
  45. 45.
    Sherali, H.D., Adams, W.P.: A hierarchy of relaxations between the continuous and convex representations for zero-one programming problems. SIAM Journal on Discrete Mathematics 3 (1990) 411–430MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Sanjeeb Dash
    • 1
  1. 1.Princeton UniversityPrincetonUSA

Personalised recommendations