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Combinatorial Bounds on Nonnegative Rank and Extended Formulations [article]

Samuel Fiorini and Volker Kaibel and Kanstantsin Pashkovich and Dirk Oliver Theis
2012 arXiv   pre-print
We study the relative power and limitations of the bounds on several examples.  ...  The main known lower bounds on the minimum sizes of extended formulations for fixed polytope P (Yannakakis 1991) are closely related to the concept of nondeterministic communication complexity.  ...  size extended formulations for the latter polytopes, most people would probably agree that for the first ones we simply still miss the right techniques to prove lower bounds on the sizes of extended formulations  ... 
arXiv:1111.0444v2 fatcat:c7buz47olnhkhfljt5uyisvsde

Combinatorial bounds on nonnegative rank and extended formulations

Samuel Fiorini, Volker Kaibel, Kanstantsin Pashkovich, Dirk Oliver Theis
2013 Discrete Mathematics  
In particular, we provide geometric interpretations (and a slight sharpening) of Yannakakis ' (1991) [35] result on the relation between minimal sizes of extended formulations and the nonnegative rank  ...  of slack matrices, and we describe the fooling set bound on the nonnegative rank (due to Dietzfelbinger et al. (1996) [7]) as the clique number of a certain graph.  ...  ones we simply still miss the right techniques to prove lower bounds on the sizes of extended formulations of concrete polytopes.  ... 
doi:10.1016/j.disc.2012.09.015 fatcat:iu3dejjm4vhmdh64ed2p6v4aoe

Extended Formulations in Combinatorial Optimization [article]

Volker Kaibel
2011 arXiv   pre-print
tools for constructing such extended formulations as well as lower bounds on their sizes.  ...  The concept of representing a polytope that is associated with some combinatorial optimization problem as a linear projection of a higher-dimensional polyhedron has recently received increasing attention  ...  Clearly, the nonnegative rank of a matrix is bounded from below by its usual rank as known from Linear Algebra.  ... 
arXiv:1104.1023v1 fatcat:zeycppq4kjggpop5q6d67t2lhm

Dagstuhl Report 13082: Communication Complexity, Linear Optimization, and lower bounds for the nonnegative rank of matrices [article]

LeRoy Beasely and Troy Lee and Hartmut Klauck and Dirk Oliver Theis
2013 arXiv   pre-print
This report documents the program and the outcomes of Dagstuhl Seminar 13082 "Communication Complexity, Linear Optimization, and lower bounds for the nonnegative rank of matrices", held in February 2013  ...  Exciting progress was reported on proving lower bounds for the nonnegative rank, on the hardness of approximation using extended formulations, and on new notions of matrix ranks.  ...  Extended Formulations and Linear Optimization Hamza Fawzi Many lower bounds on the nonnegative rank only make use of the zero/nonzero pattern of the matrix.  ... 
arXiv:1305.4147v1 fatcat:guody7xpqbdohbet6kef72u5qi

Lifts of convex sets in optimization

Volker Kaibel, Rekha Thomas
2015 Mathematical programming  
Also, the connection between lifts and nonnegative ranks have been extended to general convex sets and closed cones.  ...  This special issue is dedicated to the geometry and complexity of lifts or extended formulations of convex sets which has been an active area of research in recent years.  ...  In Lower bounds on nonnegative rank via nonnegative nuclear norms Hamza Fawzi and Pablo Parrilo describe a new lower bound on the nonnegative rank in terms of a copositive program.  ... 
doi:10.1007/s10107-015-0940-z fatcat:dojppko74nhpzjvuzbvp5gerlu

An information complexity approach to extended formulations

Mark Braverman, Ankur Moitra
2013 Proceedings of the 45th annual ACM symposium on Symposium on theory of computing - STOC '13  
Here we prove an optimal and unconditional lower bound against extended formulations for clique that matches Håstad's [16] celebrated hardness result.  ...  There has been considerable recent interest in proving lower bounds for extended formulations. Fiorini et al [14] proved that there is no polynomial sized extended formulation for traveling salesman.  ...  The nonnegative rank plays a central role in lower bounds for extended formulations, and here we will explain this connection in more detail.  ... 
doi:10.1145/2488608.2488629 dblp:conf/stoc/BravermanM13 fatcat:xoybyggzfffqffl4n2elxkrx2y

Limitations of the Hyperplane Separation Technique for Bounding the Extension Complexity of Polytopes [article]

Matthias Brugger
2020 arXiv   pre-print
We illustrate the limitations of the hyperplane separation bound, a non-combinatorial lower bound on the extension complexity of a polytope.  ...  These bounds may, however, be strengthened by normalizing rows and columns of the slack matrices.  ...  Schulz and Stefan Weltge for helpful discussions and comments. He would also like to thank an anonymous referee whose comments on an earlier version greatly improved the paper.  ... 
arXiv:1911.01541v2 fatcat:eu2a5rxavjbolmfir4m6kwb74a

Tropical lower bounds for extended formulations

Yaroslav Shitov
2014 Mathematical programming  
Swart "constructs" polynomial size extended formulations for the TSP polytope and "proves" that P = N P . 1991. Yannakakis publishes a foundational paper on extended formulations.  ...  Swart "constructs" polynomial size extended formulations for the TSP polytope and "proves" that P = N P . 1991. Yannakakis publishes a foundational paper on extended formulations.  ...  Padrol (2015) improves the lower bound wcc(n) ≥ 2 √ 2n − 2 − 1 . These lower bounds are achieved on generic (that is, random) polygons. Can we do better?  ... 
doi:10.1007/s10107-014-0833-6 fatcat:vjbr2wgsxfhjthekocbd46kiuu

On the linear extension complexity of regular n-gons

Arnaud Vandaele, Nicolas Gillis, François Glineur
2017 Linear Algebra and its Applications  
Our bounds are based on the equivalence between the computation of (i) an extended formulation of size r of a polytope P, and (ii) a rank-r nonnegative factorization of a slack matrix of the polytope P  ...  The upper bound is a slight improvement of the result of Fiorini, Rothvoss and Tiwary [Extended Formulations for Polygons, Discrete Comput. Geom. 48(3), pp. 658-668, 2012].  ...  Acknowledgement We kinldy acknowledge the participants of the Dagstuhl seminar 15082 on 'Limitations of convex programming: lower bounds on extended formulations and factorization ranks' for insightful  ... 
doi:10.1016/j.laa.2016.12.023 fatcat:lpymok2tizdi7ijtwrkleham3u

Common Information and Unique Disjointness

Gabor Braun, Sebastian Pokutta
2013 2013 IEEE 54th Annual Symposium on Foundations of Computer Science  
We provide an information-theoretic framework for establishing strong lower bounds on the nonnegative rank of matrices by means of common information, a notion previously introduced in Wyner [1975].  ...  Common information is a natural lower bound for the nonnegative rank of a matrix and by combining it with Hellinger distance estimations we compute the (almost) exact common information of UDISJ (unique  ...  Acknowledgements We are indebted to Samuel Fiorini for providing extensive feedback through the various stages of this work and who helped to significantly improve the presentation.  ... 
doi:10.1109/focs.2013.79 dblp:conf/focs/BraunP13 fatcat:7gi55ie2ofhkfeturecofpfgou

Affine reductions for LPs and SDPs [article]

Gábor Braun, Sebastian Pokutta, Daniel Zink
2016 arXiv   pre-print
We define a reduction mechanism for LP and SDP formulations that degrades approximation factors in a controlled fashion.  ...  showing an inapproximability factor of 1/2+ε for bounded degree IndependentSet.  ...  We are indebted to Siu On Chan for some of the PCP inapproximability bounds as well as Santosh Vempala for the helpful discussions.  ... 
arXiv:1410.8816v5 fatcat:tqklld6tnbbmvokxxag2l6sjem

The matching polytope does not admit fully-polynomial size relaxation schemes [chapter]

Gábor Braun, Sebastian Pokutta
2014 Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms  
While the original proof is based on the hyperplane separation bound (also called the rectangle corruption bound), we employ the information-theoretic notion of common information as introduced in Braun  ...  We generalize this result by deriving strong bounds on the polyhedral inapproximability of the matching polytope: for fixed 0 < ε < 1, every polyhedral (1 + ε / n)-approximation requires an exponential  ...  The most closely related one is providing a general information-theoretic framework for lower bounding the extension complexity of polytopes in terms of information, motivated by as well as .  ... 
doi:10.1137/1.9781611973730.57 dblp:conf/soda/BraunP15 fatcat:simbiafkvzfflnf2rc4jhgsstu

Support-based lower bounds for the positive semidefinite rank of a nonnegative matrix [article]

Troy Lee, Dirk Oliver Theis
2013 arXiv   pre-print
The most important, lower bound technique for nonnegative rank is solely based on the support of the matrix S, i.e., its zero/non-zero pattern.  ...  In this paper, we characterize the power of lower bounds on positive semidefinite rank based on solely on the support.  ...  Letchford for discussions about specific positive semidefinite formulations of various optimization problems.  ... 
arXiv:1203.3961v4 fatcat:xmzv2c6gafbazc2anuk63xm2wm

Worst-Case Results For Positive Semidefinite Rank [article]

João Gouveia, Richard Z. Robinson, Rekha R. Thomas
2014 arXiv   pre-print
Using geometry and bounds on quantifier elimination, we show that this decision can be made in polynomial time when k is fixed.  ...  In general, a nonnegative matrix of rank (k+1 choose 2) has psd rank at least k and we pose the problem of deciding whether the psd rank is exactly k.  ...  Combinatorial Bounds on Nonnegative Rank and Extended Formulations. Discrete Mathematics, 313(1):67-83, 2013. [7] S. Fiorini, S. Massar, S. Pokutta, H.R. Tiwary, and R. de Wolf.  ... 
arXiv:1305.4600v2 fatcat:ojkddhsw4zfzjmvyrtby5jxo2m

On the linear extension complexity of stable set polytopes for perfect graphs

Hao Hu, Monique Laurent
2018 European journal of combinatorics (Print)  
Exploiting the link between extension complexity and the nonnegative rank of an associated slack matrix, we investigate the behaviour of the extension complexity under these graph operations.  ...  We show bounds for the extension complexity of the stable set polytope of a perfect graph G depending linearly on the size of G and involving the depth of a decomposition tree of G in terms of basic perfect  ...  We thank Ronald de Wolf for useful discussions and comments about the topic of this paper.  ... 
doi:10.1016/j.ejc.2018.02.014 fatcat:2xt5svpcu5gotjfzygpiiq5xke
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