An extension of disjunctive programming and its impact for compact tree formulations

In the 1970's, Balas introduced the concept of disjunctive programming, which is optimization over unions of polyhedra. One main result of his theory is that, given linear descriptions for each of the polyhedra to be taken in the union, one can easily derive an extended formulation of the convex hull of the union of these polyhedra. In this paper, we give a generalization of this result by extending the polyhedral structure of the variables coupling the polyhedra taken in the union. Using this generalized concept, we derive polynomial size linear programming formulations (compact formulations) for a well-known spanning tree approximation of Steiner trees, for Gomory-Hu trees, and, as a consequence, of the minimum $T$-cut problem (but not for the associated $T$-cut polyhedron). Recently, Kaibel and Loos (2010) introduced a more involved framework called {\em polyhedral branching systems} to derive extended formulations. The most parts of our model can be expressed in terms of their framework. The value of our model can be seen in the fact that it completes their framework by an interesting algorithmic aspect.Comment: 17 page

Connected Spatial Networks over Random Points and a Route-Length Statistic

We review mathematically tractable models for connected networks on random points in the plane, emphasizing the class of proximity graphs which deserves to be better known to applied probabilists and statisticians. We introduce and motivate a particular statistic $R$ measuring shortness of routes in a network. We illustrate, via Monte Carlo in part, the trade-off between normalized network length and $R$ in a one-parameter family of proximity graphs. How close this family comes to the optimal trade-off over all possible networks remains an intriguing open question. The paper is a write-up of a talk developed by the first author during 2007--2009.Comment: Published in at http://dx.doi.org/10.1214/10-STS335 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

QoS multicast tree construction in IP/DWDM optical internet by bio-inspired algorithms

Copyright @ Elsevier Ltd. All rights reserved.In this paper, two bio-inspired Quality of Service (QoS) multicast algorithms are proposed in IP over dense wavelength division multiplexing (DWDM) optical Internet. Given a QoS multicast request and the delay interval required by the application, both algorithms are able to find a flexible QoS-based cost suboptimal routing tree. They first construct the multicast trees based on ant colony optimization and artificial immune algorithm, respectively. Then a dedicated wavelength assignment algorithm is proposed to assign wavelengths to the trees aiming to minimize the delay of the wavelength conversion. In both algorithms, multicast routing and wavelength assignment are integrated into a single process. Therefore, they can find the multicast trees on which the least wavelength conversion delay is achieved. Load balance is also considered in both algorithms. Simulation results show that these two bio-inspired algorithms can construct high performance QoS routing trees for multicast applications in IP/DWDM optical Internet.This work was supported in part ny the Program for New Century Excellent Talents in University, the Engineering and Physical Sciences Research Council (EPSRC) of UK under Grant EP/E060722/1, the National Natural Science Foundation of China under Grant no. 60673159 and 70671020, the National High-Tech Reasearch and Development Plan of China under Grant no. 2007AA041201, and the Specialized Research Fund for the Doctoral Program of Higher Education under Grant no. 20070145017

Optimal Flood Control

A mathematical model for optimal control of the water levels in a chain of reservoirs is studied. Some remarks regarding sensitivity with respect to the time horizon, terminal cost and forecast of inflow are made

Low Cost Quality of Service Multicast Routing in High Speed Networks

Many of the services envisaged for high speed networks, such as B-ISDN/ATM, will support real-time applications with large numbers of users. Examples of these types of application range from those used by closed groups, such as private video meetings or conferences, where all participants must be known to the sender, to applications used by open groups, such as video lectures, where partcipants need not be known by the sender. These types of application will require high volumes of network resources in addition to the real-time delay constraints on data delivery. For these reasons, several multicast routing heuristics have been proposed to support both interactive and distribution multimedia services, in high speed networks. The objective of such heuristics is to minimise the multicast tree cost while maintaining a real-time bound on delay. Previous evaluation work has compared the relative average performance of some of these heuristics and concludes that they are generally efficient, although some perform better for small multicast groups and others perform better for larger groups. Firstly, we present a detailed analysis and evaluation of some of these heuristics which illustrates that in some situations their average performance is reversed; a heuristic that in general produces efficient solutions for small multicasts may sometimes produce a more efficient solution for a particular large multicast, in a specific network. Also, in a limited number of cases using Dijkstra's algorithm produces the best result. We conclude that the efficiency of a heuristic solution depends on the topology of both the network and the multicast, and that it is difficult to predict. Because of this unpredictability we propose the integration of two heuristics with Dijkstra's shortest path tree algorithm to produce a hybrid that consistently generates efficient multicast solutions for all possible multicast groups in any network. These heuristics are based on Dijkstra's algorithm which maintains acceptable time complexity for the hybrid, and they rarely produce inefficient solutions for the same network/multicast. The resulting performance attained is generally good and in the rare worst cases is that of the shortest path tree. The performance of our hybrid is supported by our evaluation results. Secondly, we examine the stability of multicast trees where multicast group membership is dynamic. We conclude that, in general, the more efficient the solution of a heuristic is, the less stable the multicast tree will be as multicast group membership changes. For this reason, while the hybrid solution we propose might be suitable for use with closed user group multicasts, which are likely to be stable, we need a different approach for open user group multicasting, where group membership may be highly volatile. We propose an extension to an existing heuristic that ensures multicast tree stability where multicast group membership is dynamic. Although this extension decreases the efficiency of the heuristics solutions, its performance is significantly better than that of the worst case, a shortest path tree. Finally, we consider how we might apply the hybrid and the extended heuristic in current and future multicast routing protocols for the Internet and for ATM Networks.

Electro-spinning/netting: A strategy for the fabrication of three-dimensional polymer nano-fiber/nets.

Since 2006, a rapid development has been achieved in a subject area, so called electro-spinning/netting (ESN), which comprises the conventional electrospinning process and a unique electro-netting process. Electro-netting overcomes the bottleneck problem of electrospinning technique and provides a versatile method for generating spider-web-like nano-nets with ultrafine fiber diameter less than 20 nm. Nano-nets, supported by the conventional electrospun nanofibers in the nano-fiber/nets (NFN) membranes, exhibit numerious attractive characteristics such as extremely small diameter, high porosity, and Steiner tree network geometry, which make NFN membranes optimal candidates for many significant applications. The progress made during the last few years in the field of ESN is highlighted in this review, with particular emphasis on results obtained in the author's research units. After a brief description of the development of the electrospinning and ESN techniques, several fundamental properties of NFN nanomaterials are addressed. Subsequently, the used polymers and the state-of-the-art strategies for the controllable fabrication of NFN membranes are highlighted in terms of the ESN process. Additionally, we highlight some potential applications associated with the remarkable features of NFN nanostructure. Our discussion is concluded with some personal perspectives on the future development in which this wonderful technique could be pursued

Partitions of Minimal Length on Manifolds

We study partitions on three dimensional manifolds which minimize the total geodesic perimeter. We propose a relaxed framework based on a $\Gamma$-convergence result and we show some numerical results. We compare our results to those already present in the literature in the case of the sphere. For general surfaces we provide an optimization algorithm on meshes which can give a good approximation of the optimal cost, starting from the results obtained using the relaxed formulation