Spanning trees short or small

We study the problem of finding small trees. Classical network design problems are considered with the additional constraint that only a specified number $k$ of nodes are required to be connected in the solution. A prototypical example is the $k$MST problem in which we require a tree of minimum weight spanning at least $k$ nodes in an edge-weighted graph. We show that the $k$MST problem is NP-hard even for points in the Euclidean plane. We provide approximation algorithms with performance ratio $2\sqrt{k}$ for the general edge-weighted case and $O(k^{1/4})$ for the case of points in the plane. Polynomial-time exact solutions are also presented for the class of decomposable graphs which includes trees, series-parallel graphs, and bounded bandwidth graphs, and for points on the boundary of a convex region in the Euclidean plane. We also investigate the problem of finding short trees, and more generally, that of finding networks with minimum diameter. A simple technique is used to provide a polynomial-time solution for finding $k$-trees of minimum diameter. We identify easy and hard problems arising in finding short networks using a framework due to T. C. Hu.Comment: 27 page

Bicriteria Network Design Problems

We study a general class of bicriteria network design problems. A generic problem in this class is as follows: Given an undirected graph and two minimization objectives (under different cost functions), with a budget specified on the first, find a <subgraph \from a given subgraph-class that minimizes the second objective subject to the budget on the first. We consider three different criteria - the total edge cost, the diameter and the maximum degree of the network. Here, we present the first polynomial-time approximation algorithms for a large class of bicriteria network design problems for the above mentioned criteria. The following general types of results are presented. First, we develop a framework for bicriteria problems and their approximations. Second, when the two criteria are the same %(note that the cost functions continue to be different) we present a ``black box'' parametric search technique. This black box takes in as input an (approximation) algorithm for the unicriterion situation and generates an approximation algorithm for the bicriteria case with only a constant factor loss in the performance guarantee. Third, when the two criteria are the diameter and the total edge costs we use a cluster-based approach to devise a approximation algorithms --- the solutions output violate both the criteria by a logarithmic factor. Finally, for the class of treewidth-bounded graphs, we provide pseudopolynomial-time algorithms for a number of bicriteria problems using dynamic programming. We show how these pseudopolynomial-time algorithms can be converted to fully polynomial-time approximation schemes using a scaling technique.Comment: 24 pages 1 figur

Impact of human extraction on tropical humid forests in the Western Ghats Uttara Kannada, South India

1. Thirty strip transects of 2400 m2 each, in the evergreen forest tract of the district of Uttara Kannada in South India; were clustered into two groups with high and low levels of disturbance on the basis of the density of perennial flowering plants and the fraction of deciduous species. 2. The set of 20 transects corresponding to low disturbance localities harboured 48 ± 6 (mean ± SD) species and 694 ± 135 individuals per transect, while the other 10 transects affected by high levels of disturbance supported 36 ± 12 species and 379 ± 135 individuals. 3. Eighty-four of the total of 200 species (operational taxonomic units) were exclusive to sites of low disturbance, and 28 to those of high disturbance; 88 species were shared by sites of high and low disturbance. This number of shared species was significantly less than expected on the basis of chance alone. 4. The differences in species richness, as well as those in species turnover (0.73 ± 0.07 for high and 0.65 ± 0.01 for low disturbance sites) were significant at the 1% level, but were as expected given the lower plant densities at sites of high disturbance. 5. Lack of coppicing ability in conjunction with their use in the plywood/matchwood industry has led to the disappearance of several evergreen species such as Syzigium gardneri and Myristica malabarica at sites with high levels of disturbance. 6. With villagers concentrating on harvests of trees in the height class of 4-8 m as poles, and commercial interests mostly extracting trees > 16 m in height, there was a reduction of around 45% across all height classes between sites of low and high levels of disturbance

An improved experimental test set-up to study the performance of granular columns

This paper describes an innovative design of a newly developed large test setup for testing the performance of footings supported on soft clay reinforced with granular columns. This advanced testing method is used to examine the settlement of footings supported on granular columns. Two important features of the equipment are (a) the axial loading system which allows samples to be consolidated under Ko condition while the load is applied onto a small foundation area of the sample, and (b) a relatively large sample size of 300-mm diameter and 400-mm high. The system is also equipped with pressure cells located beneath the footing and top cap to measure the pressure distribution with respect to foundation displacement and a lateral strain gage to monitor boundary effects. This paper reports on some of the early findings from the preliminary tests carried out using this equipment. Samples for testing were prepared by consolidating kaolin slurry in a large one-dimensional consolidation chamber. The granular columns were installed using the replacement method by compacting crushed basalt (uniformly graded with 90 % between 1.5–2-mm particle sizes) into a preformed hole. The preliminary tests have yielded promising results, validating the functionality of the equipment and support the prospect of increasing the knowledge with respect to settlement response and design of a footing supported on granular columns

Analytic results for NN particles with 1/r21/r^2 interaction in two dimensions and an external magnetic field

The $2N$-dimensional quantum problem of $N$ particles (e.g. electrons) with interaction $\beta/r^2$ in a two-dimensional parabolic potential $\omega_0$ (e.g. quantum dot) and magnetic field $B$, reduces exactly to solving a $(2N-4)$-dimensional problem which is independent of $B$ and $\omega_0$. An exact, infinite set of relative mode excitations are obtained for any $N$. The $N=3$ problem reduces to that of a ficticious particle in a two-dimensional, non-linear potential of strength $\beta$, subject to a ficticious magnetic field $B_{\rm fic}\propto J$, the relative angular momentum.Comment: To appear in Physical Review Letters (in press). RevTeX file. Two figures available from [email protected] or [email protected]

Evaluating bird communities of Western Ghats to plan for a biodiversity friendly development

Reconciling development with conservation of biological diversity has emerged as a significant concern in recent years. This has been primarily attempted through establishment of protected areas taken out of mainstream development, and through regulating impacts of major development projects with the help of environmental impact assessment exercises. We believe that these two instruments need to be complemented by continually providing inputs into the biodiversity implications of ongoing development processes (and accompanying habitat transformations) at the landscape and regional level. It is desirable that such assessment of biodiversity implications is based on a transparent, objective methodology which could be used by a wide range of practitioners working with the emerging decentralized processes of development planning. In this paper we outline such a methodology focussing on birds. This involves assigning a conservation value to bird species based on readily available information on their geographical range, habitat preference, endangerment and taxonomic distinctiveness. This may then be translated into a mean composite conservation value for bird assemblages characteristic of different habitat types. By combining this information with that on ongoing processes of habitat transformations, we can provide an assessment of how development processes are affecting biodiversity values. We illustrate this methodology by assessing the conservation value of 586 bird species of Western Ghats, and a sample of bird assemblages of seven major habitat types of the region. We conclude that the most serious loss of biodiversity value arises in the transformation of montane evergreen shola forests/high altitude grasslands into monoculture plantations