We study some combinatorial and algorithmic problems related to transitions in Euclidean minimum spanning trees arising from an arbitrary motion of one or more points of the input set.3.

The paper continues the development of a primal-dual interior point algorithm for linear programming. The topics studied include simple bounds on primal variables, incorporation of Lustig's phase 1 algorithm and the use of Schur complements to handle dense columns. Extensive numerical results demonstrate the efficiency of the resulting algorithm. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

The problem of finding a minimum-weight k-connected spanning subgraph of a complete graph, assuming that the edge weights satisfy the triangle inequality, is studied. It is shown that the class of minimum-weight k-edge connected spanning subgraphs can be restricted to those subgraphs which, in addition to the connectivity requirements, satisfy the following two conditions: (I) Every vertex has degree k or $k + 1$; (II) Removing any $1, 2, \cdots ,$ or k edges does not leave the resulting connected components all k-edge connected. For the k-vertex connected case, the parallel result is obtained with “k-edge” replaced by “k-vertex,” with the added technical restriction that $| V |\geqq 2k$ for condition (I) to hold. This generalizes recent work of Monma, Munson, and Pulleyblank for the case $k = 2$.

In this paper a general integer linear programming model is presented for the important practical problem of designing minimum-cost survivable networks, and this model is related to concepts in graph theory and polyhedral combinatorics. In particular, several interesting special cases of this general model are considered, including the minimum spanning tree problem, the Steiner tree problem, and the minimum cost k-edge connected and k-node connected network design problems. The integer polyhedra associated with these problems are studied, those inequalities from natural ILP-formulations that define facets are identified, the separation problem for these facets is addressed, and how good lower bounds can be obtained from the models studied here is indicated.

The purpose of this paper is to describe in detail an implementation of a primal-dual interior point method for solving linear programming problems. Preliminary computational results indicate that this implementation compares favorably with a comparable implementation of a dual affine interior point method, and with MINOS 5.0, a state-of-the-art implementation of the simplex method. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

The authors describe methods to design cost-effective survivable telecommunications networks that use fiber-optic transmission links. One of these methods utilizes optical switching devices to implement route diversity during cable cuts. These methods have been incorporated into a software tool consisting of three modules: a topology generator, a circuit to DS3 bundler, and a multiplex layout system. This tool is compact enough to run on a personal computer. Each of these three modules is described, and sample results are provided.>

Abstract In this paper we study the problem of finding a minimum‐weight collection of edges in a planar graph which separates a given set of vertices from the outer face. This problem has two variants: either a given embedding is specified, or the best possible embedding is to be found. We present polynomial‐time algorithms for each case. We show how to use these results to recognize a special case of the steiner tree problem in graphs which is polynomially solvable. A closely related problem is shown to be NP ‐complete.

An algorithm is presented for finding a maximum-weight spanning tree of a set of n points in the Euclidean plane, where the weight of an edge (pi, pj) equals the Euclidean distance between the points pi and pj. The algorithm runs in time Ο (n logn) and requires Ο (n) space. If the points are vertices of a convex polygon (given in order along the boundary), then our algorithm requires only a linear amount of time and space. These bounds are the best possible in the algebraic computation-tree model. We also establish various properties of maximum spanning trees that can be exploited to solve other geometric problems.

The pair $(G,D)$ consisting of a planar graph $G = (V,E)$ with n vertices together with a subset of d special vertices $D \subseteq V$ is called k-planar if there is an embedding of G in the plane so that at most k faces of G are required to cover all of the vertices in D. Checking 1-planarity can be done in linear-time since it reduces to a problem of checking planarity of a related graph. We present an algorithm which given a graph G and a value k either determines that G is not k-planar or generates an appropriate embedding and associated minimum cover in $O(c^k n)$ time, where c is a constant. Hence, the algorithm runs in linear time for any fixed k. The fact that the time required by the algorithm grows exponentially in k is to be expected since we also show that for arbitrary k, the associated decision problem is strongly NP-complete, even when the planar graph has essentially a unique planar embedding, $d = \theta (n)$, and all facial cycles have bounded length. These results provide a polynomial-time recognition algorithm for special cases of Steiner tree problems in graphs which are solvable in polynomial time.

This paper describes a packet network design and analysis (PANDA) model which captures the important features of different packet technologies. This model evolved from many iterations with technology developers and network planners over several years. The main contribution is a methodology for designing low-cost backbone packet networks with satisfactory performance which is both practical and useful. This methodology is useful for investigating cost/performance tradeoffs of various network capabilities and components, thus providing a means for identifying potential cost and performance bottlenecks for different packet technologies and to guide capability requirements for new technologies.