Principally Box-integer Polyhedra and Equimodular Matrices

A polyhedron is box-integer if its intersection with any integer box $\{\ell\leq x \leq u\}$ is integer. We define principally box-integer polyhedra to be the polyhedra $P$ such that $kP$ is box-integer whenever $kP$ is integer. We characterize them in several ways, involving equimodular matrices and box-total dual integral (box-TDI) systems. A rational $r\times n$ matrix is equimodular if it has full row rank and its nonzero $r\times r$ 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 $P$ is principally box-integer. - The polyhedron $P$ is box-TDI. - Every face-defining matrix of $P$ is equimodular. - Every face of $P$ has an equimodular face-defining matrix. - Every face of $P$ has a totally unimodular face-defining matrix. - For every face $F$ of $P$, lin($F$) has a totally unimodular basis. Along our proof, we show that a cone $\{x:Ax\leq \mathbf{0}\}$ 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 $\beta$ 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, $\beta(G) \leq 1$ if and only if the matching polytope of the graph $G$ is completely described by non-negativity, star and odd-circuit inequalities. This is essentially equivalent to the h-perfection of the line-graph of $G$, as observed by Cao and Nemhauser. The complexity of computing $\beta$ is apparently not known. We show that deciding whether $\beta(G)\leq 1$ 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 $\beta$ 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 $G$ and every $c\in\mathbb{Z}_{+}^{V(G)}$, the weighted chromatic number of $(G,c)$ is the minimum cardinality of a multi-set $\mathcal{F}$ of stable sets of $G$ such that every $v\in V(G)$ belongs to at least $c_v$ members of $\mathcal{F}$. We prove that every h-perfect line-graph and every t-perfect claw-free graph $G$ has the integer round-up property for the chromatic number: for every non-negative integer weight $c$ on the vertices of $G$, the weighted chromatic number of $(G,c)$ can be obtained by rounding up its fractional relaxation. In other words, the stable set polytope of $G$ 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 $D$, let $FK(D)$ denote the fractional kernel polytope defined on $D$, and let ${\sigma}(D)$ denote the linear system defining $FK(D)$. A digraph $D$ is called kernel perfect if every induced subdigraph $D^\prime$ has a kernel, called kernel ideal if $FK(D^\prime)$ is integral for each induced subdigraph $D^\prime$, and called kernel Mengerian if ${\sigma} (D^\prime)$ is TDI for each induced subdigraph $D^\prime$. 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

tt-perfection in P5P_5-free graphs

A graph is called $t$-perfect if its stable set polytope is fully described by non-negativity, edge and odd-cycle constraints. We characterise $P_5$-free $t$-perfect graphs in terms of forbidden $t$-minors. Moreover, we show that $P_5$-free $t$-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 $4$-cycle by deletions of degree-$2$ vertices and by $t$-contractions at degree-$3$ vertices. A $t$-contraction simultaneously contracts all incident edges at a vertex with stable neighbourhood. The operation is mainly used in the field of $t$-perfect graphs. We further show that a non-bipartite quadrangulation of the projective plane can be transformed into an odd wheel by $t$-contractions and deletions of degree-$2$ vertices. We deduce that a quadrangulation of the projective plane is (strongly) $t$-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 $t$-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