# The matching polytope has exponential extension complexity

@article{Rothvoss2014TheMP, title={The matching polytope has exponential extension complexity}, author={Thomas Rothvoss}, journal={Proceedings of the forty-sixth annual ACM symposium on Theory of computing}, year={2014} }

A popular method in combinatorial optimization is to express polytopes P, which may potentially have exponentially many facets, as solutions of linear programs that use few extra variables to reduce the number of constraints down to a polynomial. After two decades of standstill, recent years have brought amazing progress in showing lower bounds for the so called extension complexity, which for a polytope P denotes the smallest number of inequalities necessary to describe a higher dimensional…

## 210 Citations

On the Extension Complexity of Stable Set Polytopes for Perfect Graphs

- Mathematics
- 2015

In linear programming one can formulate many combinatorial optimization problems as optimizing a linear function over a feasible region that is a polytope. Given a polytope P , any non-redundant…

Simple Extensions of Polytopes

- MathematicsIPCO
- 2014

A combinatorial method is devised to establish lower bounds on the simple extension complexity of a polytope P and show for several polytopes that they have large simple extension complexities.

On some problems related to 2-level polytopes

- Mathematics
- 2018

This thesis investigates a number of problems related to 2-level polytopes, in particular regarding their combinatorial structure and extension complexity, and gives an output-efficient algorithm to write down extended formulations for the stable set polytope of perfect graphs.

The Matching Problem Has No Fully Polynomial Size Linear Programming Relaxation Schemes

- MathematicsIEEE Transactions on Information Theory
- 2015

It turns out that the high extension complexity for the matching problem stems from the same source of hardness as in the case of the correlation polytope: a direct sum structure.

A geometric lower bound on the extension complexity of polytopes based on the f-vector

- Mathematics, Computer ScienceDiscret. Appl. Math.
- 2021

A generalization of extension complexity that captures P

- Mathematics, Computer ScienceInf. Process. Lett.
- 2015

Complexity of combinatorial optimization problems in terms of face lattices of associated polytopes

- Mathematics
- 2016

This paper deals with the following question: Can combinatorial properties of polytopes help in finding an estimate for the complexity of the corresponding optimization problem? Sometimes, these key…

Características de polítopos

- Computer Science, Mathematics
- 2016

This work will do the asymptotic study of two lower bounds for the extension complexity of a polytope, one combinatorial and another one geometric, and explore an upper bound technique, recovering and extending the available results.

Lower Bounds for Approximating the Matching Polytope

- Mathematics, Computer Science
- 2018

We prove that any extended formulation that approximates the matching polytope on nvertex graphs up to a factor of (1 + ε) for any 2 n ≤ ε ≤ 1 must have at least ( n α/ε ) defining inequalities where…

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