Ramified rectilinear polygons: coordinatization by dendrons

Simple rectilinear polygons (i.e. rectilinear polygons without holes or cutpoints) can be regarded as finite rectangular cell complexes coordinatized by two finite dendrons. The intrinsic $l_1$-metric is thus inherited from the product of the two finite dendrons via an isometric embedding. The rectangular cell complexes that share this same embedding property are called ramified rectilinear polygons. The links of vertices in these cell complexes may be arbitrary bipartite graphs, in contrast to simple rectilinear polygons where the links of points are either 4-cycles or paths of length at most 3. Ramified rectilinear polygons are particular instances of rectangular complexes obtained from cube-free median graphs, or equivalently simply connected rectangular complexes with triangle-free links. The underlying graphs of finite ramified rectilinear polygons can be recognized among graphs in linear time by a Lexicographic Breadth-First-Search. Whereas the symmetry of a simple rectilinear polygon is very restricted (with automorphism group being a subgroup of the dihedral group $D_4$), ramified rectilinear polygons are universal: every finite group is the automorphism group of some ramified rectilinear polygon.Comment: 27 pages, 6 figure

Optimal location of single and multiple obnoxious facilities: Algorithms for the maximin criterion under different norms.

This thesis investigates the computational problem of locating obnoxious (undesirable) facilities in a way that minimizes their effect on a given set of clients (e.g. population centres). Supposing that the undesirable effects of such a facility on a given client are a decreasing function of the distance between them the objective is to locate these facilities as far away as possible from the given set of clients, subject to constraints that prevent location at infinity. Emphasis is given to the MAXIMIN criterion which is to maximize the minimum client-to-facility distance. Distances are measured either in the Euclidean or the rectilinear metric. The properties of the optimal solution to the single facility problem are viewed from different, seemingly unrelated, perspectives ranging from plane geometry to duality theory. In particular, duality results from a mixed integer programming model are used to derive new properties of the optimal solution to the rectilinear problem. A new algorithm is developed for the rectilinear problem where the feasible region is a convex polygon. Unlike previous approaches, this method does not require linear programming at all. In addition to this, an interactive graphical approach is proposed as a site-generation tool used to identify potential locations in realistic problems. Its main advantages are that it requires minimal user intervention and makes no assumptions regarding the feasible region. It has been applied in large scale problems with up to 1000 clients, whereas the largest reported application so far involved 10 clients. Alternative models are presented for the multi-facility problem as well. Each of them is based on different assumptions and is applicable to specific situations. Moreover, an algorithm is established for the two-facility problem based on the properties of the optimal solution. To the best of our knowledge this is the first attempt to address this problem in the plane. Finally, a number of unresolved issues, especially in the multi-facility problem, are outlined and suggested as further research topics

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application

Multi-Level Multi-Objective Programming and Optimization for Integrated Air Defense System Disruption

The U.S. military\u27s ability to project military force is being challenged. This research develops and demonstrates the application of three respective sensor location, relocation, and network intrusion models to provide the mathematical basis for the strategic engagement of emerging technologically advanced, highly-mobile, Integrated Air Defense Systems. First, we propose a bilevel mathematical programming model for locating a heterogeneous set of sensors to maximize the minimum exposure of an intruder\u27s penetration path through a defended region. Next, we formulate a multi-objective, bilevel optimization model to relocate surviving sensors to maximize an intruder\u27s minimal expected exposure to traverse a defended border region, minimize the maximum sensor relocation time, and minimize the total number of sensors requiring relocation. Lastly, we present a trilevel, attacker-defender-attacker formulation for the heterogeneous sensor network intrusion problem to optimally incapacitate a subset of the defender\u27s sensors and degrade a subset of the defender\u27s network to ultimately determine the attacker\u27s optimal penetration path through a defended network