# 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}
}
• Published 3 November 2011
• Mathematics
• ArXiv
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.
216 Citations

## Figures from this paper

Efficient Protocols for Generating Bipartite Classical Distributions and Quantum States
• Computer Science
IEEE Transactions on Information Theory
• 2013
The fundamental problem of generating bipartite classical distributions or quantum states is investigated and a number of intriguing connections to fundamental measures in optimization, convex geometry, and information theory are established.
On Limits to the Scope of the Extended Formulations "Barriers"
• Mathematics
ArXiv
• 2013
The notion of augmentation for polytopes is introduced and used to show the error in two presumptions that have been key in arriving at over-reaching/over-scoped claims of "impossibility" in recent extended formulations (EF) developments.
The Complexity of Extended Formulations
Tyger Tyger, burning bright, In the forests of the night; What immortal hand or eye, Could frame thy fearful symmetry? William Blake To Mom and Dad. ACKNOWLEDGMENTS First I would like to thank my
Limits to the scope of applicability of extended formulations for LP models of combinatorial optimization problems: A summary
• Mathematics
ArXiv
• 2014
It is shown that new definitions of the notion of "projection" on which some of the recent "extended formulations" works have been based can cause those works to over-reach in their conclusions in relating polytopes to one another when the sets of the descriptive variables for those poly topes are disjoint.
Positive Semidefinite Matrix Factorization: A Connection With Phase Retrieval and Affine Rank Minimization
• Computer Science
IEEE Transactions on Signal Processing
• 2021
The results support the claim that the PSDMF framework can inherit desired numerical properties from PR and ARM algorithms, leading to more efficient PS DMF algorithms, and motivate further study of the links between these models.
Multipartite Quantum Correlation and Communication Complexities
• Computer Science
computational complexity
• 2016
This paper extends the notion of PSD-rank of matrices to that of tensors, and uses it to bound the quantum correlation complexity for generating multipartite classical distributions.
On "Exponential Lower Bounds for Polytopes in Combinatorial Optimization" by Fiorini et al. (2015): A Refutation For Models With Disjoint Sets of Descriptive Variables
• Mathematics
ArXiv
• 2016
A numerical refutation of the developments of Fiorini et al. (2015)* for models with disjoint sets of descriptive variables and an insight into the meaning of the existence of a one-to-one linear map between solutions of such models is provided.
Complexity of LP in Terms of the Face Lattice
This work considers the problem of optimizing linear function f(x) = cx on X, where c ∈ Z is an input vector and the key parameters for evaluating the complexity are the dimension d, the cardinality |X|, and the encoding size S(X) = log 2.
Formulas are Exponentially Stronger than Monotone Circuits in Non-commutative Setting
• Mathematics, Computer Science
2013 IEEE Conference on Computational Complexity
• 2013
Every monotone non-commutative circuit computing f must have an exponential size, which gives, a fortiori, an exponential separation betweenmonotone and general formulas, monot one and general branching programs, and monotones and general circuits.
Approximation Limits of Linear Programs (Beyond Hierarchies)
• Mathematics, Computer Science
2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
• 2012
A quantitative improvement of Razborov's rectangle corruption lemma for the high error regime gives strong lower bounds on the nonnegative rank of certain perturbations of the unique disjoint ness matrix that proves that quadratic approximations for CLIQUE require linear programs of exponential size.

## References

SHOWING 1-10 OF 75 REFERENCES
Extended formulations, non-negative factorizations and randomized communication
• protocols. arXiv:1105.4127,
• 2011
A counterexample to the Alon-Saks-Seymour conjecture and related problems
• Mathematics
Comb.
• 2010
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
• Computer Science, Mathematics
ISCO
• 2012
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
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.
Lifts of Convex Sets and Cone Factorizations
• Mathematics
Math. Oper. Res.
• 2013
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
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 $${2^{n/2\cdot(1-o(1))}}$$ .