Convex polyhedra of doubly stochastic matrices—IV

Basic geometrical properties of general convex polyhedra of doubly stochastic matrices are investigated. The faces of such polyhedra are characterized, and their dimensions and facets are determined. A connection between bounded faces of doubly stochastic polyhedra and faces of transportation polytopes is established, and it is shown that there exists an absolute bound for the number of extreme points of d-dimensional bounded faces of these polyhedra.     

Equifacetted 4-polytopes

Many polyhedra (convex 3-polytopes) are known which occur as facets (3-faces) of convex 4-polytopes. The purpose of this paper is to determine the combinatorial types of infinitely many more polyhedra with this property. This is achieved by determining the facets of polar bicyclic polytopes.     

Vertex-Disjoint Packing of Two Steiner Trees: polyhedra and branch-and-cut

Consider the problem of routing the electrical connections among two large terminal sets in circuit layout. A realistic model for this problem is given by the vertex-disjoint packing of two Steiner trees (2VPST), which is known to be NP-complete. This work presents an investigation on the 2VPST polyhedra. The main idea is to start from facet-defining inequalities for a vertex-weighted Steiner tree polyhedra. Some of these inequalities are proven to also define facets for the packing polyhedra, while others are lifted to derive new important families of inequalities, including proven facets. Separation algorithms are provided. Branch-and-cut implementation issues are also discussed, including some new practical techniques to improve the performance of the algorithm. The resulting code is capable of solving problems on grid graphs with up to 10000 vertices and 5000 terminals in a few minutes. Received: August 1999 / Accepted: January 2001?Published online April 12, 2001     

Binary group and Chinese postman polyhedra

A new proof of the characterization of the Chinese postman polyhedra is given. In developing this proof, a theorem of Gomory about homomorphic lifting of facets for group polyhedra is generalized to subproblems. Some results for the Chinese postman problem are generalized to binary group problems. In addition, a connection is made between Fulkerson's blocking polyhedra and a blocking pair of binary group problems. A connection is also developed between minors and lifting of facets for group problems.     

The Face Structure and Geometry of Marked Order Polyhedra

We study a class of polyhedra associated to marked posets. Examples of these polyhedra are Gelfand–Tsetlin polytopes and cones, as well as Berenstein–Zelevinsky polytopes—all of which have appeared in the representation theory of semi-simple Lie algebras. The faces of these polyhedra correspond to certain partitions of the underlying poset and we give a combinatorial characterization of these partitions. We specify a class of marked posets that give rise to polyhedra with facets in correspondence to the covering relations of the poset. On the convex geometrical side, we describe the recession cone of the polyhedra, discuss products and give a Minkowski sum decomposition. We briefly discuss intersections with affine subspaces that have also appeared in representation theory and recently in the theory of finite Hilbert space frames.     

On box totally dual integral polyhedra

Edmonds and Giles introduced the class of box totally dual integral polyhedra as a generalization of submodular flow polyhedra. In this paper a geometric characterization of these polyhedra is given. This geometric result is used to show that each TDI defining system for a box TDI polyhedron is in fact a box TDI system, that the class of box TDI polyhedra is in co-NP and is closed under taking projections and dominants, that the class of box perfect graphs is in co-NP, and a result of Edmonds and Giles which is related to the facets of box TDI polyhdera.Supported by a grant from the Alexander von Humboldt-Stiftung.     

Facets of two Steiner arborescence polyhedra

The Steiner arborescence (or Steiner directed tree) problem concerns the connection of a set of target vertices of a digraph to a given root vertex. This problem is known to be NP-hard. In the present paper we study the facial structure of two polyhedra associated with the problem. Several classes of valid inequalities are considered, and a new class with arbitrarily large coefficients is introduced. All these inequalities are shown to define distinct facets of both the Steiner polyhedra considered. This is achieved by exploiting two lifting theorems which also allow generalization of the new inequalities. Composition theorems are finally given and used to derive large families of new facet-inducing inequalities with exponentially large coefficients.     

There exist non‐isomorphic STS(19) with equivalent point codes

The rows of a point by block incidence matrix of a design can be used to generate a code, the point code of the design. It is known that binary point codes of non‐isomorphic Steiner triple systems of order hr STS(v), are inequivalent when v ≤ 15, but whether this also holds for higher orders has been open. In the current paper, an example of two non‐isomorphic STS(19) with equivalent point codes is presented. © 2004 Wiley Periodicals, Inc.     

Existence of non‐resolvable Steiner triple systems

We consider two well‐known constructions for Steiner triple systems. The first construction is recursive and uses an STS(v) to produce a non‐resolvable STS(2v + 1), for v ≡ 1 (mod 6). The other construction is the Wilson construction that we specify to give a non‐resolvable STS(v), for v ≡ 3 (mod 6), v > 9. © 2004 Wiley Periodicals, Inc. J Combin Designs 13: 16–24, 2005.     

Optimum matching forests III: Facets of matching forest polyhedra

In [3] we presented a linear system which definesP(G), the convex hull of incidence vectors of matching forests of a mixed graphG. However, many of the inequalities of this system may be redundant. Here we describe the dimension of the facets ofP(G) obtained by setting one inequality of the defining system forP(G) to an equation. This leads to a presentation of a minimal defining linear system forP(G), i.e., to a presentation of the facets ofP(G). This generalizes earlier characterizations of facets of 1-matching polyhedra and of branching polyhedra.Research partially supported by a N.R.C. Canada Postdoctorate Fellowship.     

A Theory of Nets for Polyhedra and Polytopes Related to Incidence Geometries

Our purpose is to elaborate a theory of planar nets or unfoldings for polyhedra, its generalization and extension to polytopes and to combinatorial polytopes, in terms of morphisms of geometries and the adjacency graph of facets.     

Triple arrays from difference sets

This paper addresses the question of whether triple arrays can be constructed from Youden squares developed from difference sets. We prove that if the difference set is abelian, then having ?1 as multiplier is both a necessary and sufficient condition for the construction to work. Using this, we are able to give a new infinite family of triple arrays. We use an alternative version of the construction to analyze the case of non‐abelian difference sets, for which we prove a sufficient condition for giving triple arrays. We do a computer search for such non‐abelian difference sets, but have not found any examples satisfying the given condition.     

On the strength of Gomory mixed-integer cuts as group cuts

Gomory mixed-integer (GMI) cuts generated from optimal simplex tableaus are known to be useful in solving mixed-integer programs. Further, it is well-known that GMI cuts can be derived from facets of Gomory’s master cyclic group polyhedron and its mixed-integer extension studied by Gomory and Johnson. In this paper we examine why cutting planes derived from other facets of master cyclic group polyhedra (group cuts) do not seem to be as useful when used in conjunction with GMI cuts. For many practical problem instances, we numerically show that once GMI cuts from different rows of the optimal simplex tableau are added to the formulation, all other group cuts from the same tableau rows are satisfied.     

Uniform two‐class regular partial Steiner triple systems

A 2‐class regular partial Steiner triple system is a partial Steiner triple system whose points can be partitioned into 2‐classes such that no triple is contained in either class and any two points belonging to the same class are contained in the same number of triples. It is uniform if the two classes have the same size. We provide necessary and sufficient conditions for the existence of uniform 2‐class regular partial Steiner triple systems.     

On the analytical solution of the linear‐fractional Riemann problem

The linear‐fractional problem is a generalization of the linear Riemann problem that includes the (non‐linear) factorization problem. In case of normal type it can be equivalently reduced to a family of homogeneous linear vector Riemann problems by space foliation and adequate substitutions. Moreover these are equivalent to systems of non‐homogeneous Toeplitz equations with special data. The reduced problem is solved by matrix factorization in various cases. Procedures for reduction to these cases are exposed. Various modified problems and generalizations are pointed out. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Determining All Universal Tilers

A universal tiler is a convex polyhedron whose every cross-section tiles the plane. In this paper, we introduce a slight-rotating operation for cross-sections of polyhedra. By applying the operation to suitably chosen cross-sections, we prove that a convex polyhedron is a universal tiler if and only if it is a tetrahedron or a pentahedron with parallel facets.     

Multi‐switching combination–combination synchronization of non‐identical fractional‐order chaotic systems

In this paper, multi‐switching combination–combination synchronization scheme has been investigated between a class of four non‐identical fractional‐order chaotic systems. The fractional‐order Lorenz and Chen's systems are taken as drive systems. The combination–combination of multi drive systems is then synchronized with the combination of fractional‐order Lü and Rössler chaotic systems. In multi‐switching combination–combination synchronization, the state variables of two drive systems synchronize with different state variables of two response systems simultaneously. Based on the stability of fractional‐order chaotic systems, the multi‐switching combination–combination synchronization of four fractional‐order non‐identical systems has been investigated. For the synchronization of four non‐identical fractional‐order chaotic systems, suitable controllers have been designed. Theoretical analysis and numerical results are presented to demonstrate the validity and feasibility of the applied method. Copyright © 2017 John Wiley & Sons, Ltd.     

Projection, Lifting and Extended Formulation in Integer and Combinatorial Optimization

This is an overview of the significance and main uses of projection, lifting and extended formulation in integer and combinatorial optimization. Its first two sections deal with those basic properties of projection that make it such an effective and useful bridge between problem formulations in different spaces, i.e. different sets of variables. They discuss topics like projection and restriction, the integrality-preserving property of projection, the dimension of projected polyhedra, conditions for facets of a polyhedron to project into facets of its projections, and so on. The next two sections describe the use of projection for comparing the strength of different formulations of the same problem, and for proving the integrality of polyhedra by using extended formulations or lifting. Section 5 deals with disjunctive programming, or optimization over unions of polyhedra, whose most important incarnation are mixed 0-1 programs and their partial relaxations. It discusses the compact representation of the convex hull of a union of polyhedra through extended formulation, the connection between the projection of the latter and the polar of the convex hull, as well as the sequential convexification of facial disjunctive programs, among them mixed 0-1 programs, with the related concept of disjunctive rank. Section 6 reviews lift-and-project cuts, the construction of cut generating linear programs, and techniques for lifting and for strengthening disjunctive cuts. Section 7 discusses the recently discovered possibility of solving the higher dimensional cut generating linear program without explicitly constructing it, by a sequence of properly chosen pivots in the simplex tableau of the linear programming relaxation. Finally, section 8 deals with different ways of combining cuts with branch and bound, and briefly discusses computational experience with lift-and-project cuts. This is an updated and extended version of the paper published in LNCS 2241, Springer, 2001 (as given in Balas, 2001). Research was supported by the National Science Foundation through grant #DMI-9802773 and by the Office of Naval Research through contract N00014-97-1-0196.     

A combinatorial theory of Grünbaum's new regular polyhedra,Part II: Complete enumeration

The new regular polyhedra as defined by Branko Grünbaum in 1977 (cf. [5]) are completely enumerated. By means of a theorem of Bieberbach, concerning the existence of invariant affine subspaces for discrete affine isometry groups (cf. [3], [2] or [1]) the standard crystallographic restrictions are established for the isometry groups of the non finite (Grünbaum-)polyhedra. Then, using an appropriate classification scheme which—compared with the similar, geometrically motivated scheme, used originally by Grünbaum—is suggested rather by the group theoretical investigations in [4], it turns out that the list of examples given in [5] is essentially complete except for one additional polyhedron.So altogether—up to similarity—there are two classes of planar polyhedra, each consisting of 3 individuals and each class consisting of the Petrie duals of the other class, and there are ten classes of non planar polyhedra: two mutually Petrie dual classes of finite polyhedra, each consisting of 9 individuals, two mutually Petrie dual classes of infinite polyhedra which are contained between two parallel planes with each of those two classes consisting of three one-parameter families of polyhedra, two further mutually Petrie dual classes each of which consists of three one parameter families of polyhedra whose convex span is the whole 3-space, two further mutually Petrie dual classes consisting of three individuals each of which spanE ³ and two further classes which are closed with respect to Petrie duality, each containing 3 individuals, all spanningE ³, two of which are Petrie dual to each other, the remaining one being Petrie dual to itself.In addition, a new classification scheme for regular polygons inE ⁿ is worked out in §9.