# Symmetry Matters for Sizes of Extended Formulations

@article{Kaibel2012SymmetryMF, title={Symmetry Matters for Sizes of Extended Formulations}, author={Volker Kaibel and Kanstantsin Pashkovich and Dirk Oliver Theis}, journal={SIAM J. Discret. Math.}, year={2012}, volume={26}, pages={1361-1382} }

In 1991, Yannakakis [J. Comput. System Sci., 43 (1991), pp. 441--466] proved that no symmetric extended formulation for the matching polytope of the complete graph $K_n$ with $n$ nodes has a number of variables and constraints that is bounded subexponentially in $n$. Here, symmetric means that the formulation remains invariant under all permutations of the nodes of $K_n$. It was also conjectured by Yannakakis that “asymmetry does not help much,” but no corresponding result for general extended…

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## 37 Citations

### Symmetry Matters for the Sizes of Extended Formulations

- MathematicsIPCO
- 2010

It is shown that for the polytopes associated with the matchings in Kn with $\lfloor\log n\rfloor$ edges there are non-symmetric extended formulations of polynomial size, while nevertheless no symmetric extended formulation of poynomial size exists.

### The matching polytope does not admit fully-polynomial size relaxation schemes

- Computer Science, MathematicsSODA
- 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.

### Simple extensions of polytopes

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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.

### Extended formulations, nonnegative factorizations, and randomized communication protocols

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It is shown that allowing randomization in the protocol can be crucial for obtaining small extended formulations, and it is proved that for the spanning tree and perfect matching polytopes, small variance in the Protocol forces large size in the extended formulation.

### The matching problem has no small symmetric SDP

- Computer Science, MathematicsMath. Program.
- 2017

The matching problem can be expressed compactly in a framework such as semidefinite programming (SDP) that is more powerful than linear programming but still allows efficient optimization, and any symmetric SDP for the matching problem has exponential size.

### Extended Formulation Lower Bounds for Combinatorial Optimization

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This dissertation will show that any linearprogramming relaxation for refuting random instances of constraint satisfaction problems requires super-polynomial size, and shows that any symmetric semidefinite programming relaxation for the matching problemin general graphs requires exponential size.

### The matching polytope has exponential extension complexity

- Computer ScienceSTOC
- 2014

By a known reduction this also improves the lower bound on the extension complexity for the TSP polytope from 2Ω(√n) to 2 Ω(n).

### Some 0/1 polytopes need exponential size extended formulations

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It is proved that there are 0/1 polytopes that do not admit a compact LP formulation and that for every n there is a set X such that conv(X) must have extension complexity at least $${2^{n/2\cdot(1-o(1))}}$$ .

### An information complexity approach to extended formulations

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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.

### Extended formulations in combinatorial optimization

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- 2013

This survey presents compact extended formulations for several graph problems involving cuts, trees, cycles and matchings, and for the mixing set, and presents the proof of Fiorini, Massar, Pokutta, Tiwary and de Wolf of an exponential lower bound for the cut polytope.

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