2,633 research outputs found
Principally Box-integer Polyhedra and Equimodular Matrices
A polyhedron is box-integer if its intersection with any integer box
is integer. We define principally box-integer polyhedra
to be the polyhedra such that is box-integer whenever is integer.
We characterize them in several ways, involving equimodular matrices and
box-total dual integral (box-TDI) systems. A rational matrix is
equimodular if it has full row rank and its nonzero determinants
all have the same absolute value. A face-defining matrix is a full row rank
matrix describing the affine hull of a face of the polyhedron. Box-TDI systems
are systems which yield strong min-max relations, and the underlying polyhedron
is called a box-TDI polyhedron. Our main result is that the following
statements are equivalent.
- The polyhedron is principally box-integer.
- The polyhedron is box-TDI.
- Every face-defining matrix of is equimodular.
- Every face of has an equimodular face-defining matrix.
- Every face of has a totally unimodular face-defining matrix.
- For every face of , lin() has a totally unimodular basis.
Along our proof, we show that a cone is box-TDI if
and only if it is box-integer, and that these properties are passed on to its
polar.
We illustrate the use of these characterizations by reviewing well known
results about box-TDI polyhedra. We also provide several applications. The
first one is a new perspective on the equivalence between two results about
binary clutters. Secondly, we refute a conjecture of Ding, Zang, and Zhao about
box-perfect graphs. Thirdly, we discuss connections with an abstract class of
polyhedra having the Integer Carath\'eodory Property. Finally, we characterize
the box-TDIness of the cone of conservative functions of a graph and provide a
corresponding box-TDI system
Ear-decompositions and the complexity of the matching polytope
The complexity of the matching polytope of graphs may be measured with the
maximum length of a starting sequence of odd ears in an
ear-decomposition. Indeed, a theorem of Edmonds and Pulleyblank shows that its
facets are defined by 2-connected factor-critical graphs, which have an odd
ear-decomposition (according to a theorem of Lov\'asz). In particular,
if and only if the matching polytope of the graph is
completely described by non-negativity, star and odd-circuit inequalities. This
is essentially equivalent to the h-perfection of the line-graph of , as
observed by Cao and Nemhauser.
The complexity of computing is apparently not known. We show that
deciding whether can be executed efficiently by looking at any
ear-decomposition starting with an odd circuit and performing basic modulo-2
computations. Such a greedy-approach is surprising in view of the complexity of
the problem in more special cases by Bruhn and Schaudt, and it is simpler than
using the Parity Minor Algorithm.
Our results imply a simple polynomial-time algorithm testing h-perfection in
line-graphs (deciding h-perfection is open in general). We also generalize our
approach to binary matroids and show that computing is a
Fixed-Parameter-Tractable problem (FPT)
Tropical Linear Spaces
We define tropical analogues of the notions of linear space and Plucker
coordinate and study their combinatorics. We introduce tropical analogues of
intersection and dualization and define a tropical linear space built by
repeated dualization and transverse intersection to be constructible. Our main
result that all constructible tropical linear spaces have the same f-vector and
are ``series-parallel''. We conjecture that this f-vector is maximal for all
tropical linear spaces with equality precisely for the series-parallel tropical
linear spaces. We present many partial results towards this conjecture.
In addition we relate tropical linear spaces to linear spaces defined over
power series fields and give many examples and counter-examples illustrating
aspects of this relationship. We describe a family of particularly nice
series-parallel linear spaces, which we term tree spaces, that realize the
conjectured maximal f-vector and are constructed in a manner similar to the
cyclic polytopes.Comment: 40 pages, 5 figure
Integer round-up property for the chromatic number of some h-perfect graphs
A graph is h-perfect if its stable set polytope can be completely described
by non-negativity, clique and odd-hole constraints. It is t-perfect if it
furthermore has no clique of size 4. For every graph and every
, the weighted chromatic number of is the
minimum cardinality of a multi-set of stable sets of such
that every belongs to at least members of .
We prove that every h-perfect line-graph and every t-perfect claw-free graph
has the integer round-up property for the chromatic number: for every
non-negative integer weight on the vertices of , the weighted chromatic
number of can be obtained by rounding up its fractional relaxation. In
other words, the stable set polytope of has the integer decomposition
property.
Our results imply the existence of a polynomial-time algorithm which computes
the weighted chromatic number of t-perfect claw-free graphs and h-perfect
line-graphs. Finally, they yield a new case of a conjecture of Goldberg and
Seymour on edge-colorings.Comment: 20 pages, 13 figure
On Kernel Mengerian Orientations of Line Multigraphs
We present a polyhedral description of kernels in orientations of line
multigraphs. Given a digraph , let denote the fractional kernel
polytope defined on , and let denote the linear system
defining . A digraph is called kernel perfect if every induced
subdigraph has a kernel, called kernel ideal if is
integral for each induced subdigraph , and called kernel Mengerian if
is TDI for each induced subdigraph . We show
that an orientation of a line multigraph is kernel perfect iff it is kernel
ideal iff it is kernel Mengerian. Our result strengthens the theorem of Borodin
et al. [3] on kernel perfect digraphs and generalizes the theorem of Kiraly and
Pap [7] on stable matching problem.Comment: 12 pages, corrected and slightly expanded versio
Matroids from hypersimplex splits
A class of matroids is introduced which is very large as it strictly contains
all paving matroids as special cases. As their key feature these split matroids
can be studied via techniques from polyhedral geometry. It turns out that the
structural properties of the split matroids can be exploited to obtain new
results in tropical geometry, especially on the rays of the tropical
Grassmannians.Comment: 28 pages, 2 figures, 3 tables - major revision, a wrong corollary has
been remove
-perfection in -free graphs
A graph is called -perfect if its stable set polytope is fully described
by non-negativity, edge and odd-cycle constraints. We characterise -free
-perfect graphs in terms of forbidden -minors. Moreover, we show that
-free -perfect graphs can always be coloured with three colours, and
that they can be recognised in polynomial time.Comment: Work of Holm et al of which we were not aware made some parts
unnecessary. This concerns mostly the section on near-bipartite graphs that
is much shorter no
Existence of unimodular triangulations - positive results
Unimodular triangulations of lattice polytopes arise in algebraic geometry,
commutative algebra, integer programming and, of course, combinatorics.
In this article, we review several classes of polytopes that do have
unimodular triangulations and constructions that preserve their existence.
We include, in particular, the first effective proof of the classical result
by Knudsen-Mumford-Waterman stating that every lattice polytope has a dilation
that admits a unimodular triangulation. Our proof yields an explicit (although
doubly exponential) bound for the dilation factor.Comment: 89 pages; changes from v2 and v1: the survey part has been expanded,
in particular the section on open question
Reducing quadrangulations of the sphere and the projective plane
We show that every quadrangulation of the sphere can be transformed into a
-cycle by deletions of degree- vertices and by -contractions at
degree- vertices. A -contraction simultaneously contracts all incident
edges at a vertex with stable neighbourhood. The operation is mainly used in
the field of -perfect graphs. We further show that a non-bipartite
quadrangulation of the projective plane can be transformed into an odd wheel by
-contractions and deletions of degree- vertices. We deduce that a
quadrangulation of the projective plane is (strongly) -perfect if and only
if the graph is bipartite.Comment: 10 pages, 4 figures, new results on quadrangulations of the sphere
added, old results about -perfection became corollarie
Symmetrical-geometry constructions defining helicoidal biostructures. The case of alpha-helix
The chain of algebraic geometry constructions permits to transfer from the
minimal surface with zero instability index, and from the lattice over the ring
of cyclotomic integers to the tetra-block helix. The tetra-block is the
7-vertex joining of four tetrahedra sharing common faces; it is considered as a
building unit for structures approximated by the chains of regular tetrahedra.
The minimality condition of the 7 - vertex tetrablock as a building unit is the
consequence of its unique mapping by the Klein's quartic (which is
characterized by the minimal hyperbolic Schwartz triangle) into the minimal
finite projective geometry. The topological stability of this helix provided by
the pitch to radius ratio H/R of 2{\pi}/({\tau}+1) ({\tau} is the golden
section) and by the local rotation axis order of 40/11=40exp(-H/R). These
parameters determine the helix of C{\alpha} atoms inside the alpha - helix with
the accuracy of up to 2%. They explain also the bonding relationship i -- i+4
between the i-th amide group and the (i+4)-th carbonil group of the residues in
the peptide chain and the observed value of the average segment length of the
alpha-helix which is equal to 11 residues. The tetra-block helix with the N,
C{\alpha}, C', O, H atoms in the symmetrically selected positions, determines
the structure of the alpha - helix. The proposed approach can display
adequately the symmetry of the helicoidal biopolymers.Comment: 18 pages, 5 figures, 1 tabl
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