Michel Deza

Facets for the cut cone I

We study facets of the cut coneCn, i.e., the cone of dimension 1/2n(n − 1) generated by the cuts of the complete graph onn vertices. Actually, the study of the facets of the cut cone is equivalent in some sense to the study of the facets of the cut polytope. We present several operations on facets and, in particular, a “lifting” procedure for constructing facets ofCn+1 from given facets of the lower dimensional coneCn. After reviewing hypermetric valid inequalities, we describe the new class of cycle inequalities and prove the facet property for several subclasses. The new class of parachute facets is developed and other known facets and valid inequalities are presented.

Bounds for permutation arrays

Abstract A permutation array (P.A.) defined on an r-set of symbols V is a v×r array of rows each of which is a permutation of the symbols of V and such that any two distinct rows have at most (at least) λ common column entries. We list all known bounds for such arrays and make improvements in certain cases. We consider, at length, the case when every pair of distinct rows of the P.A. have precisely λ common column entries.

Pentaheptite Modifications of the Graphite Sheet

Pentaheptites (three-coordinate tilings of the plane by pentagons and heptagons only) are classified under the chemically motivated restriction that all pentagons occur in isolated pairs and all heptagons have three heptagonal neighbors. They span a continuum between the two lattices exemplified by the boron nets in ThMoB4 (cmm) and YCrB4 (pgg), in analogy with the crossover from cubic-close-packed to hexagonal-close-packed packings in 3D. Symmetries realizable for these pentaheptite layers are three strip groups (periodic in one dimension), p1a1, p112, and p111, and five Fedorov groups (periodic in two dimensions), cmm, pgg, pg, p2, and p1. All can be constructed by simultaneous rotation of the central bonds of pyrene tilings of the graphite sheet. The unique lattice of cmm symmetry corresponds to the previously proposed pentaheptite carbon metal. Analogous pentagon-heptagon tilings on other surfaces including the torus, Klein bottle, and cylinder, face-regular tilings of pentagons and b-gons, and a full characterization of tilings involving isolated pairs and/or triples of pentagons are presented. The Kelvin paradigm of a continuum of structures arising from propagation of two original motifs has many potential applications in 2D and 3D.

The Ridge Graph of the Metric Polytope and Some Relatives

The metric polytope is a \(\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array}} \right) \) -dimensional convex polytope defined by its 4 \(\left( {\begin{array}{*{20}c} n \\ 3 \\ \end{array}} \right) \) facets. The vertices of the metric polytope are known only up to n = 6, for n = 7 they number more than 60 000. The study of the metric polytope and its relatives (the metric cone, the cut polytope and the cut cone) is mainly motivated by their application to the maximum cut and multicommodity flow feasibility problems. We characterize the ridge graph of the metric polytope, i.e. the edge graph of its dual, and, as corollary, obtain that the diameter of the dual metric polytope is 2. For n ≥ 5, the edge graph of the metric polytope restricted to its integral vertices called cuts, and to some \(\left\{ {\frac{1}{3},\,\frac{2}{3}} \right\} \) -valued vertices called anticuts, is, besides the clique on the cuts, the bipartite double of the complement of the folded n-cube. We also give similar results for the metric cone, the cut polytope and the cut cone.

Clin d'oeil on L 1 -embeddable planar graphs

In this note we present some properties of LI-embeddable planar graphs. We present a characterization of graphs isometrically embeddable into half-cubes. This result implies that every planar Li-graph G has a scale 2 embedding into a hypercube. Further, under some additional conditions we show that for a simple circuit C of a planar Li-graph G the subgraph H of G bounded by C is also Li-embeddable. In many important cases, the length of C is the dimension of the smallest cube in which H has a scale embedding. Using these facts we establish the LI-embeddability of a list of planar graphs.

The cut cone, L1 embeddability, complexity, and multicommodity flows

A finite metric (or more properly semimetric) on n points is a nonnegative vector d = (dij) 1 ⩽ i < j ⩽ n that satisfies the triangle inequality dij ⩽ dik + djk. The L1 (or Manhattan) distance ‖x − y‖1 between two vectors x = (xi) and y = (yi) in Rm is given by ‖x − y‖1 = ∑1⩽i⩽m |xi − yi|. A metric d is L1-embeddable if there exist vectors z1, z2,…, zn in Rm for some m, such that dij = ‖zi − zj‖1 for 1 ⩽ i < j ⩽ n. A cut metric is a metric with all distances zero or one and corresponds to the incidence vector of a cut in the complete graph on n vertices. The cut cone Hn is the convex cone formed by taking all nonnegative combinations of cut metrics. It is easily shown that a metric is L1-embeddable if and only if it is contained in the cut cone. In this expository paper, we provide a unified setting for describing a number of results related to L1-embeddability and the cut cone. We collect and describe results on the facial structure of the cut cone and the complexity of testing the L1-embeddability of a metric. One of the main sections of the paper describes the role of L1-embeddability in the feasibility problem for multi-commodity flows. The Ford and Fulkerson theorem for the existence of a single commodity flow can be restated as an inequality that must be valid for all cut metrics. A more general result, known as the Japanese theorem, gives a condition for the existence of a multicommodity flow. This theorem gives an inequality that must be satisfied by all metrics. For multicommodity flows involving a small number of terminals, it is known that the condition of the Japanese theorem can be replaced with one of the Ford–Fulkerson type. We review these results and show that the existence of such Ford–Fulkerson-type conditions for flows with few terminals depends critically on the fact that certain metrics are L1-embeddable.

Applications of cut polyhedra—I

Abstract In this paper and in its continuation (Part II, this issue), we group, within a unified framework, many applications of the following polyhedra: cut, boolean quadric, hypermetric and metric polyhedra. We treat, in particular, the following applications: • l1- and L1-metrics in functional analysis, • the max-cut problem, the Boole problem and multicommodity flow problems in combinatorial optimization, • lattice holes in geometry of numbers, • density matrices of many-fermions systems in quantum mechanics. We present some other applications, in probability theory, statistical data analysis and design theory. In this first part, after introducing the main definitions and operations for cut polyhedra, we describe the connections with l1-metrics and with other metric properties, and we consider, in particular, the applications to some classes of metrics arising from graphs, normed spaces and lattices.

Space fullerenes: a computer search for new Frank-Kasper structures.

A Frank-Kasper structure is a 3-periodic tiling of the Euclidean space E3 by tetrahedra such that the vertex figure of any vertex belongs to four specified patterns with, respectively, 20, 24, 26 and 28 faces. Frank-Kasper structures occur in the crystallography of metallic alloys and clathrates. A new computer enumeration method has been devised for obtaining Frank-Kasper structures of up to 20 cells in a reduced fundamental domain. Here, the 84 obtained structures have been compared with the known 27 physical structures and the known special constructions by Frank-Kasper-Sullivan, Shoemaker-Shoemaker, Sadoc-Mosseri and Deza-Shtogrin.

THE HYPERMETRIC CONE IS POLYHEDRAL

The hypermetric coneHn is the cone in the spaceRn(n−1)/2 of all vectorsd=(dij)1≤i<j≤n satisfying the hypermetric inequalities: −1≤i≤j≤nzjzjdij≤ 0 for all integer vectorsz inZn with −1≤i≤nzi=1. We explore connections of the hypermetric cone with quadratic forms and the geometry of numbers (empty spheres andL-polytopes in lattices). As an application, we show that the hypermetric coneHn is polyhedral.

Facets for the cut cone II: clique-web inequalities

We study new classes of facets for the cut coneCn generated by the cuts of the complete graph onn vertices. This cone can also be interpreted as the cone of all semi-metrics onn points that are isometricallyl1-embeddable and, in fact, the study of the facets of the cut polytope is in some sense equivalent to the study of the facets ofCn. These new facets belong to the class of clique-web inequalities which generalize the hypermetric and cycle inequalities as well as the bicycle odd wheel inequalities.