Exponential Lower Bounds on the Lengths of Some Classes of Branch-and-Cut Proofs

We examine the complexity of branch-and-cut proofs in the context of 0-1 integer programs. We establish an exponential lower bound on the length of branch-and-cut proofs that use 0-1 branching and lift-and-project cuts (called simple disjunctive cuts by some authors), Gomory-Chvatal cuts, and cuts arising from the N0 matrix-cut operator of Lovasz 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.

[1]  László Lovász,et al.  Stable sets and polynomials , 1994, Discret. Math..

[2]  Manfred Kudlek,et al.  On the Complexity of Propositional Calculus Formulae , 2001, Sci. Ann. Cuza Univ..

[3]  Alberto Caprara,et al.  On the separation of split cuts and related inequalities , 2003, Math. Program..

[4]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.

[5]  Gérard Cornuéjols,et al.  Elementary closures for integer programs , 2001, Oper. Res. Lett..

[6]  Friedrich Eisenbrand,et al.  Bounds on the Chvátal Rank of Polytopes in the 0/1-Cube , 1999, IPCO.

[7]  P. Pudlák.  Sets and Proofs: On the Complexity of the Propositional Calculus , 1999 .

[8]  Warren P. Adams,et al.  A hierarchy of relaxation between the continuous and convex hull representations , 1990 .

[9]  Michel X. Goemans,et al.  When Does the Positive Semidefiniteness Constraint Help in Lifting Procedures? , 2001, Math. Oper. Res..

[10]  Friedrich Eisenbrand,et al.  Bounds on the Chvátal Rank of Polytopes in the 0/1-Cube* , 2003, Comb..

[11]  Dima Grigoriev,et al.  Complexity of Semi-algebraic Proofs , 2002, STACS.

[12]  Pavel Pudlák,et al.  Lower bounds for resolution and cutting plane proofs and monotone computations , 1997, Journal of Symbolic Logic.

[13]  Mark Evan Hartmann,et al.  Cutting planes and the complexity of the integer hull , 1989 .

[14]  Vasek Chvátal.  Cutting Planes in Combinatorics , 1985, Eur. J. Comb..

[15]  Alexander Schrijver,et al.  On Cutting Planes , 1980 .

[16]  Ran Raz,et al.  Lower bounds for cutting planes proofs with small coefficients , 1995, Symposium on the Theory of Computing.

[17]  Miklós Ajtai,et al.  The complexity of the Pigeonhole Principle , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[18]  Stephen A. Cook,et al.  An Exponential Lower Bound for the Size of Monotone Real Circuits , 1999, J. Comput. Syst. Sci..

[19]  Vasek Chvátal,et al.  Edmonds polytopes and a hierarchy of combinatorial problems , 1973, Discret. Math..

[20]  Jan Krajícek,et al.  Lower bounds to the size of constant-depth propositional proofs , 1994, Journal of Symbolic Logic.

[21]  William J. Cook,et al.  Chvátal closures for mixed integer programming problems , 1990, Math. Program..

[22]  Robert G. Jeroslow,et al.  Trivial integer programs unsolvable by branch-and-bound , 1974, Math. Program..

[23]  William J. Cook,et al.  On cutting-plane proofs in combinatorial optimization , 1989 .

[24]  William J. Cook.  Cutting-plane proofs in polynomial space , 1990, Math. Program..

[25]  Peter Clote,et al.  Cutting planes, connectivity, and threshold logic , 1996, Arch. Math. Log..

[26]  Armin Haken,et al.  The Intractability of Resolution , 1985, Theor. Comput. Sci..

[27]  William J. Cook,et al.  On the complexity of cutting-plane proofs , 1987, Discret. Appl. Math..

[28]  Toniann Pitassi,et al.  Propositional Proof Complexity: Past, Present and Future , 2001, Bull. EATCS.

[29]  Sanjeeb Dash.  On the Matrix Cuts of Lovász and Schrijver and Their Use in Integer Programming , 2001 .

[30]  Martin W. P. Savelsbergh,et al.  Lifted Cover Inequalities for 0-1 Integer Programs: Complexity , 1999, INFORMS J. Comput..

[31]  Edward A. Hirsch,et al.  Several notes on the power of Gomory-Chvátal cuts , 2003, Ann. Pure Appl. Log..

[32]  Noga Alon,et al.  The monotone circuit complexity of boolean functions , 1987, Comb..

[33]  Jan Kra,et al.  Lower Bounds to the Size of Constant-depth Propositional Proofs , 1994 .

[34]  Jan Krajícek,et al.  Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic , 1997, Journal of Symbolic Logic.

[35]  Discretely Ordered Modules as a First-Order Extension of The Cutting Planes Proof System , 1998, J. Symb. Log..

[36]  Egon Balas,et al.  A lift-and-project cutting plane algorithm for mixed 0–1 programs , 1993, Math. Program..

[37]  Gérard Cornuéjols,et al.  A connection between cutting plane theory and the geometry of numbers , 2002, Math. Program..

[38]  Alexander Schrijver,et al.  Cones of Matrices and Set-Functions and 0-1 Optimization , 1991, SIAM J. Optim..

[39]  Laurence A. Wolsey,et al.  Aggregation and Mixed Integer Rounding to Solve MIPs , 2001, Oper. Res..

[40]  Ravi B. Boppana,et al.  The Complexity of Finite Functions , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[41]  Egon Balas,et al.  Gomory cuts revisited , 1996, Oper. Res. Lett..

[42]  Robert E. Bixby,et al.  MIP: Theory and Practice - Closing the Gap , 1999, System Modelling and Optimization.

[43]  William J. Cook,et al.  On the Matrix-Cut Rank of Polyhedra , 2001, Math. Oper. Res..

[44]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[45]  Laurence A. Wolsey,et al.  A recursive procedure to generate all cuts for 0–1 mixed integer programs , 1990, Math. Program..

[46]  Vasek Chvátal,et al.  Hard Knapsack Problems , 1980, Oper. Res..

[47]  Friedrich Eisenbrand,et al.  NOTE – On the Membership Problem for the Elementary Closure of a Polyhedron , 1999, Comb..

[48]  R. Gomory.  AN ALGORITHM FOR THE MIXED INTEGER PROBLEM , 1960 .

[49]  William J. Cook,et al.  On the Complexity of Branch and Cut Methods for the Traveling Salesman Problem , 1990, Polyhedral Combinatorics.

[50]  A. Razborov.  Unprovability of lower bounds on circuit size in certain fragments of bounded arithmetic , 1995 .

[51]  Monique Laurent,et al.  Tighter Linear and Semidefinite Relaxations for Max-Cut Based on the Lov[a-acute]sz--Schrijver Lift-and-Project Procedure , 2002, SIAM J. Optim..

[52]  Hanif D. Sherali,et al.  A Hierarchy of Relaxations Between the Continuous and Convex Hull Representations for Zero-One Programming Problems , 1990, SIAM J. Discret. Math..