Constant-Factor Approximation for TSP with Disks

We revisit the traveling salesman problem with neighborhoods (TSPN) and present the first constant-ratio approximation for disks in the plane: Given a set of $n$ disks in the plane, a TSP tour whose length is at most $O(1)$ times the optimal can be computed in time that is polynomial in $n$. Our result is the first constant-ratio approximation for a class of planar convex bodies of arbitrary size and arbitrary intersections. In order to achieve a $O(1)$-approximation, we reduce the traveling salesman problem with disks, up to constant factors, to a minimum weight hitting set problem in a geometric hypergraph. The connection between TSPN and hitting sets in geometric hypergraphs, established here, is likely to have future applications.Comment: 14 pages, 3 figure

Applications of combinatorial optimization arising from large scale surveys

Many difficult statistical problems arising in censuses or in other large scale surveys have an underlying Combinatorial Optimization structure and can be solved with Combinatorial Optimization techniques. These techniques are often more efficient than the ad hoc solution techniques already developed in the field of Statistics. This thesis considers in detail two relevant cases of such statistical problems, and proposes solution approaches based on Combinatorial Optimization and Graph Theory. The first problem is the delineation of Functional Regions, the second one concerns the selection of the scope of a large survey, as briefly described below. The purpose of this work is therefore the innovative application of known techniques to very important and economically relevant practical problems that the "Censuses, Administrative and Statistical Registers Department" (DICA) of the Italian National Institute of Statistics (Istat), where I am senior researcher, has been dealing with. In several economical, statistical and geographical applications, a territory must be partitioned into Functional Regions. This operation is called Functional Regionalization. Functional Regions are areas that typically exceed administrative boundaries, and they are of interest for the evaluation of the social and economical phenomena under analysis. Functional Regions are not fixed and politically delimited, but are determined only by the interactions among all the localities of a territory. In this thesis, we focus on interactions represented by the daily journey-to-work flows between localities in which people live and/or work. Functional Regionalization of a territory often turns out to be computationally difficult, because of the size (that is, the number of localities constituting the territory under study) and the nature of the journey-to-work matrix (that is, the sparsity). In this thesis, we propose an innovative approach to Functional Regionalization based on the solution of graph partition problems over an undirected graph called transitions graph, which is generated by using the journey-to-work data. In this approach, the problem is solved by recursively partitioning the transition graph by using the min cut algorithms proposed by Stoer and Wagner and Brinkmeier. %In the second approach, the problem is solved maximizing a function of the sizes and interactions of subsets identified by successions of partitions obtained via Multilevel partitioning approach. This approach is applied to the determination of the Functional Regions for the Italian administrative regions. The target population of a statistical survey, also called scope, is the set of statistical units that should be surveyed. In the case of some large surveys or censuses, the scope cannot be the set of all available units, but it must be selected from this set. Surveying each unit has a cost and brings a different portion of the whole information. In this thesis, we focus on the case of Agricultural Census. In this case, the units are farms, and we want to determine a subset of units producing the minimum total cost and safeguarding at least a certain portion of the total information, according to the coverage levels assigned by the European regulations. Uncertainty aspects also occur, because the portion of information corresponding to each unit is not perfectly known before surveying it. The basic decision aspect is to establish the inclusion criteria before surveying each unit. We propose here to solve the described problem using multidimensional binary knapsack models

Applications of combinatorial optimization arising from large scale surveys

Many difficult statistical problems arising in censuses or in other large scale surveys have an underlying Combinatorial Optimization structure and can be solved with Combinatorial Optimization techniques. These techniques are often more efficient than the ad hoc solution techniques already developed in the field of Statistics. This thesis considers in detail two relevant cases of such statistical problems, and proposes solution approaches based on Combinatorial Optimization and Graph Theory. The first problem is the delineation of Functional Regions, the second one concerns the selection of the scope of a large survey, as briefly described below. The purpose of this work is therefore the innovative application of known techniques to very important and economically relevant practical problems that the "Censuses, Administrative and Statistical Registers Department" (DICA) of the Italian National Institute of Statistics (Istat), where I am senior researcher, has been dealing with. In several economical, statistical and geographical applications, a territory must be partitioned into Functional Regions. This operation is called Functional Regionalization. Functional Regions are areas that typically exceed administrative boundaries, and they are of interest for the evaluation of the social and economical phenomena under analysis. Functional Regions are not fixed and politically delimited, but are determined only by the interactions among all the localities of a territory. In this thesis, we focus on interactions represented by the daily journey-to-work flows between localities in which people live and/or work. Functional Regionalization of a territory often turns out to be computationally difficult, because of the size (that is, the number of localities constituting the territory under study) and the nature of the journey-to-work matrix (that is, the sparsity). In this thesis, we propose an innovative approach to Functional Regionalization based on the solution of graph partition problems over an undirected graph called transitions graph, which is generated by using the journey-to-work data. In this approach, the problem is solved by recursively partitioning the transition graph by using the min cut algorithms proposed by Stoer and Wagner and Brinkmeier. %In the second approach, the problem is solved maximizing a function of the sizes and interactions of subsets identified by successions of partitions obtained via Multilevel partitioning approach. This approach is applied to the determination of the Functional Regions for the Italian administrative regions. The target population of a statistical survey, also called scope, is the set of statistical units that should be surveyed. In the case of some large surveys or censuses, the scope cannot be the set of all available units, but it must be selected from this set. Surveying each unit has a cost and brings a different portion of the whole information. In this thesis, we focus on the case of Agricultural Census. In this case, the units are farms, and we want to determine a subset of units producing the minimum total cost and safeguarding at least a certain portion of the total information, according to the coverage levels assigned by the European regulations. Uncertainty aspects also occur, because the portion of information corresponding to each unit is not perfectly known before surveying it. The basic decision aspect is to establish the inclusion criteria before surveying each unit. We propose here to solve the described problem using multidimensional binary knapsack models

A PTAS for Euclidean TSP with Hyperplane Neighborhoods

In the Traveling Salesperson Problem with Neighborhoods (TSPN), we are given a collection of geometric regions in some space. The goal is to output a tour of minimum length that visits at least one point in each region. Even in the Euclidean plane, TSPN is known to be APX-hard, which gives rise to studying more tractable special cases of the problem. In this paper, we focus on the fundamental special case of regions that are hyperplanes in the $d$-dimensional Euclidean space. This case contrasts the much-better understood case of so-called fat regions. While for $d=2$ an exact algorithm with running time $O(n^5)$ is known, settling the exact approximability of the problem for $d=3$ has been repeatedly posed as an open question. To date, only an approximation algorithm with guarantee exponential in $d$ is known, and NP-hardness remains open. For arbitrary fixed $d$, we develop a Polynomial Time Approximation Scheme (PTAS) that works for both the tour and path version of the problem. Our algorithm is based on approximating the convex hull of the optimal tour by a convex polytope of bounded complexity. Such polytopes are represented as solutions of a sophisticated LP formulation, which we combine with the enumeration of crucial properties of the tour. As the approximation guarantee approaches $1$, our scheme adjusts the complexity of the considered polytopes accordingly. In the analysis of our approximation scheme, we show that our search space includes a sufficiently good approximation of the optimum. To do so, we develop a novel and general sparsification technique to transform an arbitrary convex polytope into one with a constant number of vertices and, in turn, into one of bounded complexity in the above sense. Hereby, we maintain important properties of the polytope

Information Extraction and Modeling from Remote Sensing Images: Application to the Enhancement of Digital Elevation Models

To deal with high complexity data such as remote sensing images presenting metric resolution over large areas, an innovative, fast and robust image processing system is presented. The modeling of increasing level of information is used to extract, represent and link image features to semantic content. The potential of the proposed techniques is demonstrated with an application to enhance and regularize digital elevation models based on information collected from RS images

A spatial decision support system for autodistricting collection units for the taking of the Canadian census

This dissertation documents the districting requirements for collection units for taking the Canadian Census and provides a spatial decision support system for their automatic creation. In the context of the literature on autodistricting, this problem falls under the general category of creating districts for monitoring, surveillance and inventory applications since the Census is essentially an spatial inventory exercise. The basic requirement is to create an area-based categorical coverage such that the workload is equitably distributed amongst Census Representatives within the limits of a large number of constraints and conditions.A new omnibus automated districting process that combines a 3-stage cascading selection procedure for identifying sub-blockface, blockface and block level collection units with a 4- stage heuristic solution procedure for grouping blocks (termed 'assigns', 'annexes', 're-assigns' and 'adjusts') is contributed by this research to provide a systematic response to varying districting situations.The resulting spatial decision support system for autodistricting has been tested on test data sets and on one of the larger urban population centres of Canada. The set of test pattern sites mimicking typical settlement patterns was generated to ensure that the various alternative assignment or block grouping methods (i.e., unidirectional and bidirectional tessellations based on circular and rectangular grids and regular, random and 'extrema-based' seeds) performed as designed and specified. The Census Subdivision of Laval (in the Census Metropolitan Area of Montreal) was selected as the test site for comparing the performance of the autodistricting capacity to the actual, manually created, results from the 1986 Census.To permit the comparison of results from classical manual and automated processes, a set of satisficing evaluation functions that vary in accordance with data availability was implemented in the context of a competing set of districting objectives. The most sophisticated of these evaluation functions incorporates a composite index that combines the distribution and a measure of the 'density' of the dwellings with the length of the route that must be followed to complete the collection activity (including travel time to the start of the route and between route parts).To assess the continued acceptability of the districting from the previous Census, and/or to select between alternative results generated by computer-assisted approaches, a set of objective functions is provided that vary depending upon the available amount of geographic, cartographic or statistical data