Linear vs. semidefinite extended formulations: exponential separation and strong lower bounds

@article{Fiorini2012LinearVS,
  title={Linear vs. semidefinite extended formulations: exponential separation and strong lower bounds},
  author={Samuel Fiorini and Serge Massar and Sebastian Pokutta and Hans Raj Tiwary and Ronald de Wolf},
  journal={ArXiv},
  year={2012},
  volume={abs/1111.0837}
}
We solve a 20-year old problem posed by Yannakakis and prove that there exists no polynomial-size linear program (LP) whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to be symmetric. Moreover, we prove that this holds also for the cut polytope and the stable set polytope. These results were discovered through a new connection that we make between one-way quantum communication protocols and semidefinite programming reformulations of LPs. 
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References

SHOWING 1-10 OF 75 REFERENCES
On the Distributional Complexity of Disjointness
Expressing combinatorial optimization problems by Linear Programs
Extended formulations, non-negative factorizations and randomized communication
  • protocols. arXiv:1105.4127,
  • 2011
A counterexample to the Alon-Saks-Seymour conjecture and related problems
TLDR
A counterexample to the conjecture that k complete bipartite graphs have chromatic number at most k+1 is constructed and several related problems in combinatorial geometry and communication complexity are discussed.
Geometry of cuts and metrics, volume 15 of Algorithms and Combinatorics
  • 1997
Extended Formulations, Nonnegative Factorizations, and Randomized Communication Protocols
We show that the binary logarithm of the nonnegative rank of a nonnegative matrix is, up to small constants, equal to the minimum complexity of a randomized communication protocol computing the…
Nondeterministic Quantum Query and Communication Complexities
TLDR
The nondeterministic quantum algorithms for Boolean functions f have positive acceptance probability on input x iff f(x)=1, which implies that the quantum communication complexities of the equality and disjointness functions are n+1 if the authors do not allow any error probability.
Combinatorial bounds on nonnegative rank and extended formulations
Lifts of Convex Sets and Cone Factorizations
TLDR
This paper addresses the basic geometric question of when a given convex set is the image under a linear map of an affine slice of a given closed convex cone and shows that the existence of a lift of a conveX set to a cone is equivalent to theexistence of a factorization of an operator associated to the set and its polar via elements in the cone and its dual.
Some 0/1 polytopes need exponential size extended formulations
TLDR
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 $${2^{n/2\cdot(1-o(1))}}$$ .
...
1
2
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5
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