On the Nearest Neighbor Rule for the Metric Traveling Salesman Problem

We present a very simple family of traveling salesman instances with $n$ cities where the nearest neighbor rule may produce a tour that is $\Theta(\log n)$ times longer than an optimum solution. Our family works for the graphic, the euclidean, and the rectilinear traveling salesman problem at the same time. It improves the so far best known lower bound in the euclidean case and proves for the first time a lower bound in the rectilinear case

Online Graph Exploration on Trees, Unicyclic Graphs and Cactus Graphs

We study the problem of exploring all vertices of an undirected weighted graph that is initially unknown to the searcher. An edge of the graph is only revealed when the searcher visits one of its endpoints. Beginning at some start node, the searcher's goal is to visit every vertex of the graph before returning to the start node on a tour as short as possible. We prove that the Nearest Neighbor algorithm's competitive ratio on trees with $n$ vertices is $\Theta(\log n)$, i.e. no better than on general graphs. Furthermore, we examine the algorithm Blocking for a range of parameters not considered previously and prove it is 3-competitive on unicyclic graphs as well as $5/2+\sqrt{2}\approx 3.91$-competitive on cactus graphs. The best known lower bound for these two graph classes is 2.Comment: 10 pages, 5 figure

Improved Lower Bound for Competitive Graph Exploration

We give an improved lower bound of 10/3 on the competitive ratio for the exploration of an undirected, edge-weighted graph with a single agent that needs to return to the starting location after visiting all vertices. We assume that the agent has full knowledge of all edges incident to visited vertices, and, in particular, vertices have unique identifiers. Our bound improves a lower bound of 2.5 by Dobrev et al. [SIROCCO'12] and also holds for planar graphs, where it complements an upper bound of 16 by Kalyanasundaram and Pruhs[TCS'94]. The question whether a constant competitive ratio can be achieved in general remains open

Self-Evaluation Applied Mathematics 2003-2008 University of Twente

This report contains the self-study for the research assessment of the Department of Applied Mathematics (AM) of the Faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS) at the University of Twente (UT). The report provides the information for the Research Assessment Committee for Applied Mathematics, dealing with mathematical sciences at the three universities of technology in the Netherlands. It describes the state of affairs pertaining to the period 1 January 2003 to 31 December 2008

Aplicação de métodos heurísticos no planeamento de rotas : o caso da Tecniwood-Soluções

Dissertação de mestrado integrado em Engenharia e Gestão IndustrialO facto de a concorrência aumentar de dia para dia leva a que as empresas tenham necessidade de se tornarem cada vez mais eficientes. Tendo em conta que a logística é uma das principais fontes de despesa de uma empresa é por isso importante que esta funcione da melhor maneira possível. A logística encontra-se presente em áreas como o transporte, controlo de inventário, compras, armazenamento, movimentação de materiais, entre outros, contudo este projeto apenas irá analisar a vertente do transporte. A presente dissertação foi desenvolvida na Tecniwood-Soluções, uma empresa de distribuição de derivados de madeira e madeira maciça, cuja logística de transporte representa uma grande fonte de despesa. Por esse motivo, é proposto o desenvolvimento de um modelo que consiga lidar com todas as restrições de uma empresa deste ramo e ao mesmo tempo consiga apresentar uma boa solução num curto período de tempo. Esse modelo, heurística do cluster mais próximo, foi confrontado com um exemplo real de um dia de planeamento de rotas da Tecniwood-Soluções, conseguindo no final apresentar uma boa solução num curto período de tempo. No futuro ainda será necessário concluir as etapas em falta na heurística do cluster mais próximo e no software que irá incorporar esse modelo.With the rise in competition among companies, all of them are increasingly obligated to become more efficient, and because logistics is one of the major sources of expenses in a company, it’s extremely important that this sector functions in the most efficient manner possible. Logistics involves different resources such as transportation, inventory, purchasing, warehousing, material handling, and many others, but this project will analyze only the transportation sector. The present dissertation was developed at Tecniwood-Soluções, which is a wood-based products and solid-wood wholesale company, where logistics is one of the major sources of expenditure. Therefore, the development of an algorithm that can deal with the constraints of a company in this field and provide, at the same time, an effective solution in a short period, was proposed. This algorithm, the nearest cluster algorithm, was tested with a real routing problem that occurs at Tecniwood-Soluções and, as a result, it was possible to develop an effective solution in a minimum of time. In the future, it will be necessary to complete the missing steps of the nearest cluster algorithm, as well as the software that will incorporate that algorithm