Lower bounds on the sizes of integer programs without additional variables

For a given set X ⊆ ℤ d of integer points, we investigate the smallest number of facets of any polyhedron whose set of integer points is conv(X) ∩ ℤ d . This quantity, which we call the relaxation complexity of X , corresponds to the smallest number of linear inequalities of any integer program having X as the set of feasible solutions that does not use auxiliary variables. We show that the use of auxiliary variables is essential for constructing polynomial size integer programming formulations in many relevant cases. In particular, we provide asymptotically tight exponential lower bounds on the relaxation complexity of the integer points of several well-known combinatorial polytopes, including the traveling salesman polytope and the spanning tree polytope.

[1]  Hans Raj Tiwary,et al.  Exponential Lower Bounds for Polytopes in Combinatorial Optimization , 2011, J. ACM.

[2]  Gérard Cornuéjols,et al.  The traveling salesman problem on a graph and some related integer polyhedra , 1985, Math. Program..

[3]  Stephen C. Graves,et al.  The Travelling Salesman Problem and Related Problems , 1978 .

[4]  Michel X. Goemans,et al.  Smallest compact formulation for the permutahedron , 2015, Math. Program..

[5]  Robert G. Jeroslow.  On defining sets of vertices of the hypercube by linear inequalities , 1975, Discret. Math..

[6]  Wojciech Szatzschneider,et al.  An Inequality , 1991, SIAM Rev..

[7]  H. Minkowski,et al.  Geometrie der Zahlen , 1896 .

[8]  Sebastian Pokutta,et al.  A note on the extension complexity of the knapsack polytope , 2013, Oper. Res. Lett..

[9]  Hans Raj Tiwary,et al.  On the extension complexity of combinatorial polytopes , 2013, Math. Program..

[10]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

[11]  R. A. Zemlin,et al.  Integer Programming Formulation of Traveling Salesman Problems , 1960, JACM.

[12]  R. Kipp Martin,et al.  Using separation algorithms to generate mixed integer model reformulations , 1991, Oper. Res. Lett..

[13]  Thomas Rothvoß,et al.  The matching polytope has exponential extension complexity , 2013, STOC.

[14]  Volker Kaibel,et al.  Lower bounds on the sizes of integer programs without additional variables , 2015, Math. Program..

[15]  Sanjeev Arora,et al.  Computational Complexity: A Modern Approach , 2009 .

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

[17]  M. Yannakakis.  Expressing combinatorial optimization problems by linear programs , 1991, Symposium on the Theory of Computing.

[18]  François Margot,et al.  On a Binary-Encoded ILP Coloring Formulation , 2007, INFORMS J. Comput..