# 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}
}
• T. Rothvoss
• Published 11 November 2013
• Computer Science
• Proceedings of the forty-sixth annual ACM symposium on Theory of computing
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

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## References

SHOWING 1-10 OF 36 REFERENCES
The matching polytope does not admit fully-polynomial size relaxation schemes
• Computer Science, Mathematics
SODA
• 2015
It turns out that the high extension complexity for the matchingpolytope stem from the same source of hardness as for the correlation polytope: a direct sum structure.
Extended Formulations for Polygons
• Mathematics
Discret. Comput. Geom.
• 2012
It is proved that there exist n-gons whose vertices lie on an O(n)×O(n2) integer grid with extension complexity $\varOmega (\sqrt{n}/\ sqrt{\log n})$.
On the Extension Complexity of Combinatorial Polytopes
• Mathematics
ICALP
• 2013
A lifting argument is described to show exponential extension complexity for a number of NP-complete problems including subset-sum and three dimensional matching and a relationship is obtained between the extension complexity of the cut polytope of a graph and that of its graph minors.
Expressing combinatorial optimization problems by linear programs
It is shown that expressing the matching and the Traveling Salesman Problem by a symmetric linear program requires exponential size, and the minimum size needed by a LP to express a polytope to a combinatorial parameter is related.
Lower Bounds on the Size of Semidefinite Programming Relaxations
• Mathematics, Computer Science
STOC
• 2015
It is proved that SDPs of polynomial-size are equivalent in power to those arising from degree-O(1) sum-of-squares relaxations, and this result yields the first super-polynomial lower bounds on the semidefinite extension complexity of any explicit family of polytopes.
An information complexity approach to extended formulations
• Computer Science
STOC '13
• 2013
An optimal and unconditional lower bound against linear programs for clique that matches Hastad's celebrated hardness result is proved and an information theoretic framework is developed to approach these questions and is used to prove the main result.
On the existence of 0/1 polytopes with high semidefinite extension complexity
• Mathematics
Math. Program.
• 2015
It is shown that there is a 0/1 polytope such that any spectrahedron projecting to it must be the intersection of a semidefinite cone of dimension $$2^{\varOmega (n)}$$2Ω(n) and an affine space.
Symmetry Matters for Sizes of Extended Formulations
• Mathematics
SIAM J. Discret. Math.
• 2012
It is shown that for the polytopes associated with the matchings in the complete graph K_n with $\lfloor\log n\rfloor$ edges there are nonsymmetric extended formulations of polynomial size, while nevertheless no symmetric extended formulation of polynnomial size exist.