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On the cycle polytope of a binary matroid

F Barahona, M Grötschel
1986 Journal of combinatorial theory. Series B (Print)  
This matroid fi is binary. A set C E E in a binary matroid i@ is called a cycle if either C = 0 or C is the disjoint union of circuits.  ...  Thus P(M, 0) can be viewed as the convex hull of the incidence vectors of the cycles of k. In fact, many different matrices may lead to one and the same binary matroid &?  ... 
doi:10.1016/0095-8956(86)90063-8 fatcat:dpzuqsmxenauhehby46v4wv5v4

Master polytopes for cycles of binary matroids

M. Grötschel, K. Truemper
1989 Linear Algebra and its Applications  
Prior work on the cycle polytopes P(M) of binary matroids M has almost exclusively concentrated on regular matroids.  ...  We show that the facets of the cycle polytopes P(L,) have a rather simple description which may be used to deduce easily some, and in principle all, facets of the cycle polytopes of general binary matroids  ...  THE CYCLE POLYTOPES OF COMPLETE BINARY MATROIDS Let L,, k > 1, be the complete binary matroids of the Introduction, i.e., L, is the matroid consisting of just one loop, and L,, k > 2, is the binary matroid  ... 
doi:10.1016/0024-3795(89)90478-3 fatcat:bipgg2dkovgjtarsojptu6gu5q

On Vertices and Facets of Combinatorial 2-Level Polytopes [chapter]

Manuel Aprile, Alfonso Cevallos, Yuri Faenza
2016 Lecture Notes in Computer Science  
We investigate upper bounds on the product of the number of facets f d−1 (P ) and the number of vertices f0(P ), where d is the dimension of a 2-level polytope P .  ...  of matroids.  ...  This statement and one of its proofs generalizes to the cycle space (the set of all cycles) of binary matroids. Lemma 23. Let M be a binary matroid with d elements and rank r.  ... 
doi:10.1007/978-3-319-45587-7_16 fatcat:4lq5xfvmbbhuzjmkwdscbqtlwy

Cycle Bases for Lattices of Binary Matroids with No Fano Dual Minor and Their One-Element Extensions

Tamás Fleiner, Winfried Hochstättler, Monique Laurent, Martin Loebl
1999 Journal of combinatorial theory. Series B (Print)  
In this paper we study the question of existence of a cycle basis (that is, a basis consisting only of cycles) for the lattice Z(M) generated by the cycles of a binary matroid M.  ...  0 of such matroid M, any cycle basis for Z(M) can be completed to a cycle basis for Z(M 0 ).  ...  The authors are grateful to Jim Geelen and Lex Schrijver for fruitful discussions.  ... 
doi:10.1006/jctb.1999.1904 fatcat:ubrsyzzoanhmtngbq7b3dv3gba

The even and odd cut polytopes

Michel Deza, Monique Laurent
1993 Discrete Mathematics  
The vertex sets of both polytopes P, and EvP, come from the cycle sets of some binary matroids; a more general example is the even T-cut polytope.  ...  Laurent, The even and odd cut polytopes, Discrete Mathematics 119 (1993) 49966. The cut polytope P, is the convex hull of the incidence vectors of all cuts of the complete graph K, on n nodes.  ...  Acknowledgements We are very grateful to Komei Fukuda who kindly provided us the full description obtained by computer of the even cut polytope on 8 nodes, as well as several additional computations.  ... 
doi:10.1016/0012-365x(93)90116-b fatcat:pglloy6t4rfa3fwfx7a3d5eu3q

Page 789 of Mathematical Reviews Vol. , Issue 90B [page]

1990 Mathematical Reviews  
Two procedures for deducing facets of the cycle polytopes of arbitrary binary matroids from the facets of the complete binary ones are described.  ...  (D-AGSB); Truemper, K. (1-TXD) Master polytopes for cycles of binary matroids. Linear Algebra Appl. 114(115) (1989), 523-540.  ... 

On 2-level polytopes arising in combinatorial settings [article]

Manuel Aprile, Alfonso Cevallos, Yuri Faenza
2017 arXiv   pre-print
affine projections of certain order polytopes; and a linear-size description of the base polytope of matroids that are 2-level in terms of cuts of an associated tree.  ...  The key to most of our proofs is a deeper understanding of the relations among those polytopes and their underlying combinatorial structures.  ...  We moreover thank one of them for an argument that significantly simplified the proof of Lemma 26 and Theorem 29.  ... 
arXiv:1702.03187v2 fatcat:fh4lhd6f65hdtnxmanqm6osppy

Page 592 of Mathematical Reviews Vol. , Issue 93b [page]

1993 Mathematical Reviews  
When MMP is in fact an MIP or a GMP, the FMMP polytope coincides with the matroid intersection polytope or with the classic fractional matching polytope of the graph, respectively.  ...  The author shows that the representation theory of Brylawski and Lucas (1973) may be extended to a representation theory for binary matroids by unitary rings rather than fields.  ... 

A new semidefinite programming hierarchy for cycles in binary matroids and cuts in graphs

João Gouveia, Monique Laurent, Pablo A. Parrilo, Rekha Thomas
2010 Mathematical programming  
In this paper we construct the theta bodies of the vanishing ideal of cycles in a binary matroid.  ...  If the binary matroid avoids certain minors we can characterize when the first theta body in the hierarchy equals the cycle polytope of the matroid.  ...  The cycle ideal of a binary matroid and its theta bodies Let M = (E, C) be a binary matroid; that is, E is a finite set and C is a collection of subsets of E that is closed under taking symmetric differences  ... 
doi:10.1007/s10107-010-0425-z fatcat:xpm65vet5zfl7g4dffhd3tt7qm

Decomposition and optimization over cycles in binary matroids

M Grötschel, K Truemper
1989 Journal of combinatorial theory. Series B (Print)  
We call this problem the maximum weight cycle problem, or just the cycle problem of binary matroids.  ...  and the Eulerian subgraph problem (if A4 is the graphic matroid of a graph G, then the cycles of M are the (not necessarily connected) Eulerian subgraphs of G).  ...  Received October 9, 1986 For k = 2 and 3, we define several k-sums of binary matroids and of polytopes arising from cycles of binary matroids.  ... 
doi:10.1016/0095-8956(89)90052-x fatcat:5dk7tccsengrvppgtiipir5d7q

Regular matroids have polynomial extension complexity [article]

Manuel Aprile, Samuel Fiorini
2019 arXiv   pre-print
Past results of Wong and Martin on extended formulations of the spanning tree polytope of a graph imply a O(n^2) bound for the special case of (co)graphic matroids.  ...  We prove that the extension complexity of the independence polytope of every regular matroid on n elements is O(n^6).  ...  (If M is a binary matroid represented by matrix A ∈ F m×n 2 , then the cycles of M are all solutions x ∈ F n 2 of Ax = 0.) Let M 1 , M 2 be binary matroids.  ... 
arXiv:1909.08539v2 fatcat:kzinxvnyqjb6ngitz5n5iqhs6y

Ear-decompositions and the complexity of the matching polytope [article]

Yohann Benchetrit, András Sebő
2015 arXiv   pre-print
We also generalize our approach to binary matroids and show that computing β is a Fixed-Parameter-Tractable problem (FPT).  ...  The complexity of the matching polytope of graphs may be measured with the maximum length β of a starting sequence of odd ears in an ear-decomposition.  ...  It is well-known that, as for graphs, the set of (incidence vectors of) cycles of a binary matroid M with ground set S is a subspace of F S 2 (this actually characterizes binary matroids [32, chap. 9 9  ... 
arXiv:1509.05586v1 fatcat:k2r7cuqplnhzhao3lug2tdneqe

The Weak-Map Order and Polytopal Decompositions of Matroid Base Polytopes [article]

Kenji Kashiwabara
2012 arXiv   pre-print
The weak-map order on the matroid base polytopes is the partial order defined by inclusion. Lucas proved that the base polytope of no binary matroid includes the base polytope of a connected matroid.  ...  base polytope should be a facet of the former matroid base polytope.  ...  If the graph has a 3-cycle on {x, y, z}, the matroid base system is 2-decomposable by ({x, y, z}, 2) = . Proof. Let {x, y, z} have such a 3-cycle.  ... 
arXiv:1201.4662v2 fatcat:2wjggshq7zd6pod73dyqqvrxwy

Extended Formulations for Independence Polytopes of Regular Matroids

Volker Kaibel, Jon Lee, Matthias Walter, Stefan Weltge
2016 Graphs and Combinatorics  
This generalizes a similar statement for (co-)graphic matroids, which is a simple consequence of Martin's extended formulation for the spanning-tree polytope.  ...  We show that the independence polytope of every regular matroid has an extended formulation of size quadratic in the size of its ground set.  ...  We would like to thank Klaus Truemper for valuable comments on the decomposition of matroids.  ... 
doi:10.1007/s00373-016-1709-8 fatcat:d2cwuz537zhhhkqwe2sfufklla

Page 1997 of Mathematical Reviews Vol. , Issue 90D [page]

1990 Mathematical Reviews  
A cycle cover of a binary matroid M is a family S of cycles of M such that every element of M belongs to at least one cycle of S.  ...  Finally it is shown that s(C,) =n-2"-'/(2" —1), where C,, is the class of binary matroids which have a cycle cover consisting of at most n cycles.  ... 
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