A Constant Factor Approximation for Capacitated Min-Max Tree Cover

Given a graph G = (V,E) with non-negative real edge lengths and an integer parameter k, the (uncapacitated) Min-Max Tree Cover problem seeks to find a set of at most k trees which together span V and each tree is a subgraph of G. The objective is to minimize the maximum length among all the trees. In this paper, we consider a capacitated generalization of the above and give the first constant factor approximation algorithm. In the capacitated version, there is a hard uniform capacity (?) on the number of vertices a tree can cover. Our result extends to the rooted version of the problem, where we are given a set of k root vertices, R and each of the covering trees is required to include a distinct vertex in R as the root. Prior to our work, the only result known was a (2k-1)-approximation algorithm for the special case when the total number of vertices in the graph is k? [Guttmann-Beck and Hassin, J. of Algorithms, 1997]. Our technique circumvents the difficulty of using the minimum spanning tree of the graph as a lower bound, which is standard for the uncapacitated version of the problem [Even et al.,OR Letters 2004] [Khani et al.,Algorithmica 2010]. Instead, we use Steiner trees that cover ? vertices along with an iterative refinement procedure that ensures that the output trees have low cost and the vertices are well distributed among the trees

On the (parameterized) complexity of recognizing well-covered (r,l)-graphs.

An (r,ℓ)(r,ℓ)-partition of a graph G is a partition of its vertex set into r independent sets and ℓℓ cliques. A graph is (r,ℓ)(r,ℓ) if it admits an (r,ℓ)(r,ℓ)-partition. A graph is well-covered if every maximal independent set is also maximum. A graph is (r,ℓ)(r,ℓ)-well-covered if it is both (r,ℓ)(r,ℓ) and well-covered. In this paper we consider two different decision problems. In the (r,ℓ)(r,ℓ)-Well-Covered Graph problem ((r,ℓ)(r,ℓ) wcg for short), we are given a graph G, and the question is whether G is an (r,ℓ)(r,ℓ)-well-covered graph. In the Well-Covered (r,ℓ)(r,ℓ)-Graph problem (wc (r,ℓ)(r,ℓ) g for short), we are given an (r,ℓ)(r,ℓ)-graph G together with an (r,ℓ)(r,ℓ)-partition of V(G) into r independent sets and ℓℓ cliques, and the question is whether G is well-covered. We classify most of these problems into P, coNP-complete, NP-complete, NP-hard, or coNP-hard. Only the cases wc(r, 0)g for r≥3r≥3 remain open. In addition, we consider the parameterized complexity of these problems for several choices of parameters, such as the size αα of a maximum independent set of the input graph, its neighborhood diversity, or the number ℓℓ of cliques in an (r,ℓ)(r,ℓ)-partition. In particular, we show that the parameterized problem of deciding whether a general graph is well-covered parameterized by αα can be reduced to the wc (0,ℓ)(0,ℓ) g problem parameterized by ℓℓ, and we prove that this latter problem is in XP but does not admit polynomial kernels unless coNP⊆NP/polycoNP⊆NP/poly

The UNOR 40 plan (1971-1972) by Hestnes Ferreira: As a more structured expansion proposal for a planning unit in Lisbon

The aim of this paper is to present the work of Hestnes Ferreira and his team, namely for the UNOR 40 planning unit in Lisbon, as a study case of an infrastructural enhancement in Mainland Portugal during the early 1970s. The UNOR design teams were recruited outside the municipal staff. For UNOR40 the team was coordinated by Raúl Hestnes Ferreira and included architects Rodrigo Rau and Vicente Bravo, landscape architect Gonçalo Ribeiro Teles, and urban geographer, Jorge Gaspar. These oversaw the planning of a large area between Campo Grande and Benfica, using a traffic study developed by French consultants. The main results of the UNOR 40 Plan were to redefine the layout of the North-South Hub, Combatentes and Lusíada Avenues, as a way of ordering the urban network of this sector, including the urban access to Telheiras. The plan also comprised the creation of an institutional square, based on a program that included museums, institutes, office buildings, and a church. However, the applicability of the UNOR 40 Plan was practically nil, with the exception of the layout of some road links.info:eu-repo/semantics/publishedVersio

Algorithms for Order-Preserving Matching

String matching is a widely studied problem in Computer Science. There have been many recent developments in this field. One fascinating problem considered lately is the order-preserving matching (OPM) problem. The task is to find all the substrings in the text which have the same length and relative order as the pattern, where the relative order is the numerical order of the numbers in a string. The problem finds its applications in the areas involving time series or series of numbers. More specifically, it is useful for those who are interested in the relative order of the pattern and not in the pattern itself. For example, it can be used by analysts in a stock market to study movements of prices.  In addition to the OPM problem, we also studied its approximate variation. In approximate order-preserving matching, we search for those substrings in the text which have relative order similar to the pattern, i.e., relative order of the pattern matches with at most k mismatches. With respect to applications of order-preserving matching, approximate search is more meaningful than exact search. We developed various advanced solutions for the problem and its variant. Special emphasis was laid on the practical efficiency of the solutions. Particularly, we introduced a simple solution for the OPM problem using filtration. We proved experimentally that our method was effective and faster than the previous solutions for the problem. In addition, we combined the Single Instruction Multiple Data (SIMD) instruction set architecture with filtration to develop competent solutions which were faster than our previous solution. Moreover, we proposed another efficient solution without filtration using the SIMD architecture. We also presented an offline solution based on the FM-index scheme. Furthermore, we proposed practical solutions for the approximate order-preserving matching problem and one of the solutions was the first sublinear solution on average for the problem

Q(sqrt(-3))-Integral Points on a Mordell Curve

We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4

Shortest Route at Dynamic Location with Node Combination-Dijkstra Algorithm

Abstract— Online transportation has become a basic requirement of the general public in support of all activities to go to work, school or vacation to the sights. Public transportation services compete to provide the best service so that consumers feel comfortable using the services offered, so that all activities are noticed, one of them is the search for the shortest route in picking the buyer or delivering to the destination. Node Combination method can minimize memory usage and this methode is more optimal when compared to A* and Ant Colony in the shortest route search like Dijkstra algorithm, but can’t store the history node that has been passed. Therefore, using node combination algorithm is very good in searching the shortest distance is not the shortest route. This paper is structured to modify the node combination algorithm to solve the problem of finding the shortest route at the dynamic location obtained from the transport fleet by displaying the nodes that have the shortest distance and will be implemented in the geographic information system in the form of map to facilitate the use of the system. Keywords— Shortest Path, Algorithm Dijkstra, Node Combination, Dynamic Location (key words