Extension Complexity of Polytopes

2020 ◽  
pp. 210-238
Keyword(s):  
Extension Complexity

scholarly journals On the linear extension complexity of regular n-gons

2017 ◽  
Vol 521 ◽  
pp. 217-239 ◽  
Author(s):  
Arnaud Vandaele ◽  
Nicolas Gillis ◽  
François Glineur
Keyword(s):  
Linear Extension ◽  
Extension Complexity

scholarly journals On the extension complexity of combinatorial polytopes

2014 ◽  
Vol 153 (1) ◽  
pp. 95-115 ◽  
Author(s):  
David Avis ◽  
Hans Raj Tiwary
Keyword(s):  
Extension Complexity

scholarly journals Smaller Extended Formulations for Spanning Tree Polytopes in Minor-closed Classes and Beyond

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Manuel Aprile ◽  
Samuel Fiorini ◽  
Tony Huynh ◽  
Gwenaël Joret ◽  
David R. Wood
Keyword(s):  
Spanning Tree ◽  
Short Proof ◽  
Induced Subgraphs ◽  
Direct Construction ◽  
Extension Complexity ◽  
Graph Classes ◽  
Closed Class ◽  
Vertex Graph ◽  
Graph Class ◽  
Strongly Sublinear

Let $G$ be a connected $n$-vertex graph in a proper minor-closed class $\mathcal G$. We prove that the extension complexity of the spanning tree polytope of $G$ is $O(n^{3/2})$. This improves on the $O(n^2)$ bounds following from the work of Wong (1980) and Martin (1991). It also extends a result of Fiorini, Huynh, Joret, and Pashkovich (2017), who obtained a $O(n^{3/2})$ bound for graphs embedded in a fixed surface. Our proof works more generally for all graph classes admitting strongly sublinear balanced separators: We prove that for every constant $\beta$ with $0<\beta<1$, if $\mathcal G$ is a graph class closed under induced subgraphs such that all $n$-vertex graphs in $\mathcal G$ have balanced separators of size $O(n^\beta)$, then the extension complexity of the spanning tree polytope of every connected $n$-vertex graph in $\mathcal{G}$ is $O(n^{1+\beta})$. We in fact give two proofs of this result, one is a direct construction of the extended formulation, the other is via communication protocols. Using the latter approach we also give a short proof of the $O(n)$ bound for planar graphs due to Williams (2002).

scholarly journals Regular Matroids Have Polynomial Extension Complexity

2021 ◽  
Author(s):  
Manuel Aprile ◽  
Samuel Fiorini
Keyword(s):  
Spanning Tree ◽  
Extended Formulations ◽  
Polynomial Extension ◽  
Regular Matroid ◽  
Extension Complexity ◽  
Special Case

We prove that the extension complexity of the independence polytope of every regular matroid on [Formula: see text] elements is [Formula: see text]. Past results of Wong and Martin on extended formulations of the spanning tree polytope of a graph imply a [Formula: see text] bound for the special case of (co)graphic matroids. However, the case of a general regular matroid was open, despite recent attempts. We also consider the extension complexity of circuit dominants of regular matroids, for which we give a [Formula: see text] bound.

scholarly journals Extension Complexity and Realization Spaces of Hypersimplices

2017 ◽  
Vol 59 (3) ◽  
pp. 621-642
Author(s):  
Francesco Grande ◽  
Arnau Padrol ◽  
Raman Sanyal
Keyword(s):  
Extension Complexity ◽  
Realization Spaces
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