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On the cycle polytope of a binary matroid

1986
*
Journal of combinatorial theory. Series B (Print)
*

This

doi:10.1016/0095-8956(86)90063-8
fatcat:dpzuqsmxenauhehby46v4wv5v4
*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*&? ...##
###
Master polytopes for cycles of binary matroids

1989
*
Linear Algebra and its Applications
*

Prior work

doi:10.1016/0024-3795(89)90478-3
fatcat:bipgg2dkovgjtarsojptu6gu5q
*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*...##
###
On Vertices and Facets of Combinatorial 2-Level Polytopes
[chapter]

2016
*
Lecture Notes in Computer Science
*

We investigate upper bounds

doi:10.1007/978-3-319-45587-7_16
fatcat:4lq5xfvmbbhuzjmkwdscbqtlwy
*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. ...##
###
Cycle Bases for Lattices of Binary Matroids with No Fano Dual Minor and Their One-Element Extensions

1999
*
Journal of combinatorial theory. Series B (Print)
*

In this paper we study

doi:10.1006/jctb.1999.1904
fatcat:ubrsyzzoanhmtngbq7b3dv3gba
*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. ...##
###
The even and odd cut polytopes

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. ...

##
###
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]

2017
*
arXiv
*
pre-print

affine projections

arXiv:1702.03187v2
fatcat:fh4lhd6f65hdtnxmanqm6osppy
*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. ...##
###
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

2010
*
Mathematical programming
*

In this paper we construct

doi:10.1007/s10107-010-0425-z
fatcat:xpm65vet5zfl7g4dffhd3tt7qm
*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 ...##
###
Decomposition and optimization over cycles in binary matroids

1989
*
Journal of combinatorial theory. Series B (Print)
*

We call this problem

doi:10.1016/0095-8956(89)90052-x
fatcat:5dk7tccsengrvppgtiipir5d7q
*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*. ...##
###
Regular matroids have polynomial extension complexity
[article]

2019
*
arXiv
*
pre-print

Past results

arXiv:1909.08539v2
fatcat:kzinxvnyqjb6ngitz5n5iqhs6y
*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*. ...##
###
Ear-decompositions and the complexity of the matching polytope
[article]

2015
*
arXiv
*
pre-print

We also generalize our approach to

arXiv:1509.05586v1
fatcat:k2r7cuqplnhzhao3lug2tdneqe
*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 ...##
###
The Weak-Map Order and Polytopal Decompositions of Matroid Base Polytopes
[article]

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*. ...

##
###
Extended Formulations for Independence Polytopes of Regular Matroids

2016
*
Graphs and Combinatorics
*

This generalizes

doi:10.1007/s00373-016-1709-8
fatcat:d2cwuz537zhhhkqwe2sfufklla
*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*. ...##
###
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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