Maximum gradient embeddings and monotone clustering

Let (X,d_X) be an n-point metric space. We show that there exists a distribution D over non-contractive embeddings into trees f:X-->T such that for every x in X, the expectation with respect to D of the maximum over y in X of the ratio d_T(f(x),f(y)) / d_X(x,y) is at most C (log n)^2, where C is a universal constant. Conversely we show that the above quadratic dependence on log n cannot be improved in general. Such embeddings, which we call maximum gradient embeddings, yield a framework for the design of approximation algorithms for a wide range of clustering problems with monotone costs, including fault-tolerant versions of k-median and facility location.Comment: 25 pages, 2 figures. Final version, minor revision of the previous one. To appear in "Combinatorica

The Non-Uniform k-Center Problem

In this paper, we introduce and study the Non-Uniform k-Center problem (NUkC). Given a finite metric space $(X,d)$ and a collection of balls of radii $\{r_1\geq \cdots \ge r_k\}$, the NUkC problem is to find a placement of their centers on the metric space and find the minimum dilation $\alpha$, such that the union of balls of radius $\alpha\cdot r_i$ around the $i$th center covers all the points in $X$. This problem naturally arises as a min-max vehicle routing problem with fleets of different speeds. The NUkC problem generalizes the classic $k$-center problem when all the $k$ radii are the same (which can be assumed to be $1$ after scaling). It also generalizes the $k$-center with outliers (kCwO) problem when there are $k$ balls of radius $1$ and $\ell$ balls of radius $0$. There are $2$-approximation and $3$-approximation algorithms known for these problems respectively; the former is best possible unless P=NP and the latter remains unimproved for 15 years. We first observe that no $O(1)$-approximation is to the optimal dilation is possible unless P=NP, implying that the NUkC problem is more non-trivial than the above two problems. Our main algorithmic result is an $(O(1),O(1))$-bi-criteria approximation result: we give an $O(1)$-approximation to the optimal dilation, however, we may open $\Theta(1)$ centers of each radii. Our techniques also allow us to prove a simple (uni-criteria), optimal $2$-approximation to the kCwO problem improving upon the long-standing $3$-factor. Our main technical contribution is a connection between the NUkC problem and the so-called firefighter problems on trees which have been studied recently in the TCS community.Comment: Adjusted the figur

Robust mean absolute deviation problems on networks with linear vertex weights

This article deals with incorporating the mean absolute deviation objective function in several robust single facility location models on networks with dynamic evolution of node weights, which are modeled by means of linear functions of a parameter. Specifically, we have considered two robustness criteria applied to the mean absolute deviation problem: the MinMax criterion, and the MinMax regret criterion. For solving the corresponding optimization problems, exact algorithms have been proposed and their complexities have been also analyzed.Ministerio de Ciencia e Innovación MTM2007-67433-C02-(01,02)Ministerio de Ciencia e Innovación MTM2009-14243Ministerio de Ciencia e Innovación MTM2010-19576-C02-(01,02)Ministerio de Ciencia e Innovación DE2009-0057Junta de Andalucía P09-TEP-5022Junta de Andalucía FQM-584

A proposal for an ecological park towards a sustainable humane habitat in Abu Qir, Alexandria, Egypt

This paper represents a preliminary stage of a multiphase research-project, carried out at the Arab Academy for Science and Technology (AAST) by the Environmental Design Research Group (EDRG) looking at alternative proposals for urban development strategies at Abu Qir Area in Alexandria, Egypt. Abu Qir is located east of Alexandria has a very important productive bay on the Mediterranean Sea and its eastern and south east coast is the only mirror of Egypt on the Dead Sea. Besides being one of the most important spots on Alexandria bay, Abu Qir is considered one of the highest polluted areas in Alexandria, although the area itself has a very high potential on the Mediterranean and the Dead Sea. This paper explores the design and development of an ecological park as a part of the urban development at Abu Qir bay. The paper investigates the proposals of conceptual design strategies for an Ecological Park at Abu-Qir. Lessons learnt from examples that have successfully introduced and developed the ideas of Ecological parks in the international built environment are reviewed. Parks are key elements in the urban ecosystem, serving dual roles as core zones of urban nature and as multi-use recreational open space. Moreover, as part of an open space system, they play important functional roles as part of landscape-scale habitat and hydrological networks. Today, there is an urging need for the development of urban parks. As urban populations have rapidly increased through civilization and industrialization, cities have been enlarged, nature has been impaired, and human relations have been damaged. For these reasons, people have lost the opportunity to come into contact with nature. Environmental problems such as air and water pollution have resulted, in conjunction with a gradual decrease in natural surroundings. Ecological destruction has caused the landscape to be devastated. For this reason, ecological parks, places serving to reintroduce beautiful birds and insects into the devastated landscape, are needed. There is a need for ecological parks as places where sound ecological values can be instilled and established through the observation and study of nature. Higher priority is given to the ecological environment than in other parks. The design proposal of the ecological park is a place that is ecologically restored and preserved for the purposes of scientific observation and study, and also to provide an easily accessible place for visitors to observe plants, animals and insects living in their natural environment. The proposed park also plays an important role as an educational open classroom for the community and aims at raising global awareness issues in Abu-Qir, thus ensuring that children grow up with an awareness of the global problems facing the world today, and how they can contribute, as active members of the society, in reducing the consequences of pollution and global warming we have created with our own hands. The paper analyses the Abu Qir village potentials and constraints and put forward an urban development strategy based on this analysis. The paper then presents the strategy that has been established to thoroughly preserve, protect and maintain areas of rich natural resources at Abu Qir, and method have been developed to allow enjoyment of them with minimum artificial influence. Eco-tours have been developed for observation of ecosystems. A variety of opportunities to experience nature has been made available by minimization of human interference and by the introduction of an "Environment Commentator" as will be discussed in this paper

Evolutionary multi-stage financial scenario tree generation

Multi-stage financial decision optimization under uncertainty depends on a careful numerical approximation of the underlying stochastic process, which describes the future returns of the selected assets or asset categories. Various approaches towards an optimal generation of discrete-time, discrete-state approximations (represented as scenario trees) have been suggested in the literature. In this paper, a new evolutionary algorithm to create scenario trees for multi-stage financial optimization models will be presented. Numerical results and implementation details conclude the paper

Unreliable point facility location problems on networks

In this paper we study facility location problems on graphs under the most common criteria, such as, median, center and centdian, but we incorporate in the objective function some reliability aspects. Assuming that facilities may become unavailable with a certain probability, the problem consists of locating facilities minimizing the overall or the maximum expected service cost in the long run, or a convex combination of the two. We show that the k-facility problem on general networks is NP-hard. Then, we provide efficient algorithms for these problems for the cases of k = 1, 2, both on general networks and on trees. We also explain how our methodology extends to handle a more general class of unreliable point facility location problems related to the ordered median objective function.Ministerio de Ciencia y TecnologíaJunta de Andalucí