### Lifting Linear Extension Complexity Bounds to the Mixed-Integer Setting

Mixed-integer mathematical programs are among the most commonly used models for a wide set of problems in Operations Research and related fields. However, there is still very little known about what can be expressed by small mixed-integer programs. In particular, prior to this work, it was open whether some classical problems, like the minimum odd-cut problem, can be expressed by a compact mixed-integer program with few (even constantly many) integer variables. This is in stark contrast to linear formulations, where recent breakthroughs in the field of extended formulations have shown that many polytopes associated to classical combinatorial optimization problems do not even admit approximate extended formulations of sub-exponential size. We provide a general framework for lifting inapproximability results of extended formulations to the setting of mixed-integer extended formulations, and obtain almost tight lower bounds on the number of integer variables needed to describe a variety of classical combinatorial optimization problems. Among the implications we obtain, we show that any mixed-integer extended formulation of sub-exponential size for the matching polytope, cut polytope, traveling salesman polytope or dominant of the odd-cut polytope, needs $\Omega(n/\log n)$ many integer variables, where $n$ is the number of vertices of the underlying graph. Conversely, the above-mentioned polyhedra admit polynomial-size mixed-integer formulations with only $O(n)$ or $O(n \log n)$ (for the traveling salesman polytope) many integer variables. Our results build upon a new decomposition technique that, for any convex set $C$, allows for approximating any mixed-integer description of $C$ by the intersection of $C$ with the union of a small number of affine subspaces.

[1]  Thomas Rothvoß The matching polytope has exponential extension complexity , 2014, STOC.

[2]  Jack Edmonds,et al.  Maximum matching and a polyhedron with 0,1-vertices , 1965 .

[3]  W. M. B. Dukes Bounds on the number of generalized partitions and some applications , 2003, Australas. J Comb..

[4]  H. P. Williams,et al.  A Survey of Different Integer Programming Formulations of the Travelling Salesman Problem , 2007 .

[5]  Benny Sudakov,et al.  Submodular Minimization Under Congruency Constraints , 2019, Comb..

[6]  Volker Kaibel,et al.  A Short Proof that the Extension Complexity of the Correlation Polytope Grows Exponentially , 2015, Discret. Comput. Geom..

[7]  Temel Öncan,et al.  A comparative analysis of several asymmetric traveling salesman problem formulations , 2009, Comput. Oper. Res..

[8]  Volker Kaibel,et al.  Maximum semidefinite and linear extension complexity of families of polytopes , 2018, Math. Program..

[9]  Egon Balas Disjunctive Programming , 2010, 50 Years of Integer Programming.

[10]  Hendrik W. Lenstra,et al.  Integer Programming with a Fixed Number of Variables , 1983, Math. Oper. Res..

[11]  Thomas Rothvoß Some 0/1 polytopes need exponential size extended formulations , 2013, Math. Program..

[12]  Alexander Barvinok,et al.  A course in convexity , 2002, Graduate studies in mathematics.

[13]  Ting-Yi Sung,et al.  An analytical comparison of different formulations of the travelling salesman problem , 1991, Math. Program..

[14]  M. Yannakakis Expressing combinatorial optimization problems by Linear Programs , 1991 .

[15]  Rahul Jain,et al.  Extension Complexity of Independent Set Polytopes , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[16]  Sebastian Pokutta,et al.  The matching polytope does not admit fully-polynomial size relaxation schemes , 2015, SODA.

[17]  G. C. Shephard Inequalities between mixed volumes of convex sets , 1960 .

[18]  Sergey I. Veselov,et al.  Integer program with bimodular matrix , 2009, Discret. Optim..

[19]  R. Tyrrell Rockafellar,et al.  Convex Analysis , 1970, Princeton Landmarks in Mathematics and Physics.

[20]  Samuel Fiorini,et al.  Approximation Limits of Linear Programs (Beyond Hierarchies) , 2015, Math. Oper. Res..

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

[22]  S. Voß,et al.  A classification of formulations for the (time-dependent) traveling salesman problem , 1995 .

[23]  Michele Conforti,et al.  Subgraph polytopes and independence polytopes of count matroids , 2015, Oper. Res. Lett..

[24]  Rico Zenklusen,et al.  Extension Complexity Lower Bounds for Mixed-Integer Extended Formulations , 2017, SODA.

[25]  Hans Raj Tiwary,et al.  On the Extension Complexity of Combinatorial Polytopes , 2013, ICALP.

[26]  Rico Zenklusen,et al.  A strongly polynomial algorithm for bimodular integer linear programming , 2017, STOC.

[27]  David K. Smith Theory of Linear and Integer Programming , 1987 .

[28]  Volker Kaibel,et al.  Lower Bounds on the Sizes of Integer Programs without Additional Variables , 2014, IPCO.

[29]  Caterina De Simone,et al.  The cut polytope and the Boolean quadric polytope , 1990, Discret. Math..

[30]  Gérard Cornuéjols,et al.  Extended formulations in combinatorial optimization , 2013, Ann. Oper. Res..