Label-connected graphs and the gossip problem

A graph with m edges is called label-connected if the edges can be labeled with real numbers in such a way that, for every pair (u, v) of vertices, there is a (u, v)-path with ascending labels. The minimum number of edges of a label-connected graph on n vertices equals the minimum number of calls in the gossip problem for n persons, which is known to be 2n − 4 for n ≥ 4. A polynomial characterization of label-connected graphs with n vertices and 2n − 4 edges is obtained. For a graph G, let θ(G) denote the minimum number of edges that have to be added to E(G) in order to create a graph with two edge-disjoint spanning trees. It is shown that for a graph G to be label-connected, θ(G) ≤ 2 is necessary and θ(G) ≤ 1 is sufficient. For i = 1, 2, the condition θ(G) ≤ i can be checked in polynomial time. Yet recognizing label-connected graphs is an NP-complete problem. This is established by first showing that the following problem is NP-complete: Given a graph G and two vertices u and v of G, does there exist a (u, v)-path P in G such that G−E(P) is connected

Service scheduling in garden maintenance

Neoturf is a Portuguese company working in the area of project, building and garden’s maintenance. Neoturf would like to have a procedure for scheduling and routing efficiently the clients from garden maintenance services. The company has two teams available during the whole year and an additional team during summer to handle all the maintenance jobs. Each team consists of two or three employees with a vehicle fully equipped with the tools that allow to carry out every kind of maintenance service. In the beginning of each year, the number and frequency of maintenance interventions to conduct during the year, on each client, are accorded. Each client is assigned to the same team and, usually, time windows are established so that visits to the client should occur only within these periods. As the Neoturf costumers’ are geographically spread over a wide region, the total distance on visiting clients is a factor that has a heavy weight on the costs of the company. Neoturf is concerned with reducing these costs, while satisfying the agreements with the clients

Warehouse Storing and Collecting of Parts

This report deals with reducing the high costs resulting from the wear and tear of the fork-lifts used to store or collect items in a warehouse. Two problems were identified and addressed separately. One concerns the way items should be stored or collected at storage locations on the shelves of one corridor. The other problem seeks for an efficient way to define which fork-lift should operate on each corridor, and the order by which the fork-lifts should visit the corridors. We give to both problems formulations that fit in the framework of combinatorial optimization

Report on "Scheduling in a factory"

In order to carry out their orders of shoe soles, this company has a number of tasks T_1, ..., T_n of different lengths to be assigned to groups of machines. Each group is operated by one worker (two in one case), and an operation cycle corresponds to injection, cooling, and removal of the sole. The time taken at each step varies from one order to another, and when starting a new task a machine needs to be tuned, which takes some extra time. Machines are working in parallel. At the moment the assignment is carried out empirically, and the problem proposed is to optimize the procedure

Injective edge coloring of graphs

Three edges $e_{1}, e_{2}$ and $e_{3}$ in a graph $G$ are consecutive if they form a path (in this order) or a cycle of lengths three. An injective edge coloring of a graph $G = (V,E)$ is a coloring $c$ of the edges of $G$ such that if $e_{1}, e_{2}$ and $e_{3}$ are consecutive edges in $G$, then $c(e_{1})\neq c(e_3)$. The injective edge coloring number $\chi_{i}^{'}(G)$ is the minimum number of colors permitted in such a coloring. In this paper, exact values of $\chi_{i}^{'}(G)$ for several classes of graphs are obtained, upper and lower bounds for $\chi_{i}^{'}(G)$ are introduced and it is proven that checking whether $\chi_{i}^{'}(G)= k$ is NP-complete.in publicatio

Linking habitats for multiple species

a b s t r a c t The establishment of linkages between habitats is of great importance to avert the detrimental impacts of land fragmentation and climate change on biodiversity. Linkages need to be cost-efficient, and should account for specific dispersal requirements of species. Since cost-efficient linkages defined independently for each individual species are more costly than linkages optimised for multiple species, there is need for methods specifically designed to retrieve efficient linkages for multiple species. MulTyLink (Multiple Type Linkages) is a Cþþ open source program that defines cost-efficient linkages free of barriers for the species considered, and that allows species-specific dispersal requirements to be considered. Here we present, discuss and illustrate the algorithms used by MulTyLink to identify costefficient linkages for multiple species

Definition of the productivity regions

Eucalyptus productivity is strongly related with climate and soil types of the area where it is planted. To accurately assess the potential productivity of that species in Portugal, plantations were monitored at different management units compartments (MUC) at several locations all over Portugal. Certain indices of productivity of the Eucalyptus at each MUC were recorded, as well as the type of climate and soil characteristics of the region. Both climate and soil, factors that affects Eucalyptus grow, were classified in in ten classes 1, 2, . . . , 10 of expected growing productivity for the Euca- lyptus. Thus, every MUC belongs to a unique pair (c, s), with 1 \leq c \leq 10 and 1 \leq s \leq 10 indicating the type of climate and the type of soil of the region where MUC is located, respectively, and it is expected that to have high (low) productivity indices when c and s are both close to 10 (1). The aim of this work is to identify regions that have similar productivity levels based on the classifications of soil and climate types and to check if the available data provided by RAIZ show that those factors affect the productivity indices. During the 5-days ESGI this team worked on the datasets provided by RAIZ, presented an update for the existing MAI productivity chart and developed new clusters for the Density, Yeld and Consumption productivity indices. The definition of some quality measures for the clusters, allowed to compare the different approaches and also point out some fragilities on the datasets. Indeed, a review of classification regarding climate and/or soil characteristics is suggested, as well as the need of a bigger sample for the Density, Yeld and Consumption productivity indices in order to get more reliable outputs

Revisiting the minimum set cover, the maximal coverage problems and a maximum benefit area selection problem to make climate‐change‐concerned conservation plans effective

1. Informed decisions for the selection of protected areas (PAs) are grounded in two general problems in Operations Research: the minimum set covering problem (minCost), where a set of ecological constraints are established as conservation targets and the minimum cost PAs are found, and the maximal coverage problem (maxCoverage) where the constraint is uniquely economic (i.e., a fixed budget) and the goal is to maximize the number of species having conservation targets adequately covered. 2. We adjust minCost and maxCoverage to accommodate the dynamic effects of climate change on species’ ranges. The selection of sites is replaced by the selection of time-ordered sequences of sites (climate change corridors), and an estimate of the persistence of each species in corridors is calculated according to the expected suitability of each site in the respective time period and the capacity of species to disperse between consecutive sites along corridors. In these problems, conservation targets are expressed as desired (and attainable) species persistence levels. We also introduce a novel problem (minShortfall) that combines minCost and maxCoverage. Unlike these two problems, minShortfall allows persistence targets to be missed and minimizes the sum of those gaps (i.e., target shortfalls), subject to a limited budget. 3. We illustrate the three problems with a case study using climatic suitability estimates for ten mammal species in the Iberian Peninsula under a climate change scenario until 2080. We compare solutions of the three problems with respect to species persistence and PA costs, under distinct settings of persistence targets, number of target-fulfilled species, and budgets. The solutions from different problems differed with regard to the areas to prioritize, their timings and the species whose persistence targets were fulfilled. This analysis also allowed identifying groups of species sharing corridors in optimal solutions, thus allowing important financial savings in site protection. 4. We suggest that enhancing species persistence is an adequate approach to cope with habitat shifts due to climate change. We trust the three problems discussed can provide complementary and valuable support for planners to anticipate decisions in order that the negative effects of climate change on species’ persistence are minimized