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Publication# Problèmes de cheminements optimaux dans les réseaux avec contraintes associées aux arcs

Thèse EPFL

Résumé

In this thesis we consider two optimal routing problems in networks. Both are NP-hard and are often used to modelize real life problems of large size. We first present the capacitated arc routing problem (CARP). In this problem, we have to serve a set of customers represented by the arcs of a network with vehicles of restricted capacity. The aim is to design a set of vehicle routes of least total cost. We describe two heuristic methods for the CARP based on local search techniques. The first one is a Tabu Search heuristic using an original post-optimization procedure. Principles appearing in this procedure are used to define a class of neighborhoods we combine in a Variable Neighborhood Descent method. Our algorithms compare favorably with some of her methods. They were initially developed to deal with the undirected CARP. We describe how to transform them so that they can be applied to the directed CARP. We also propose an adaptation to the directed case of a lower bound originally developed for the undirected CARP. This lower bound is used to evaluate the quality of the solutions produced by our algorithms for the directed CARP. Our methods obtain solutions whose value is, on average, close to known lower bounds. We use the CARP to tackle a real life problem of boat routing and scheduling. This problem can be viewed as a directed CARP with time windows. We describe in detail its modeling process and we analyze our first results. We obtain solutions that require less boats than the present schedule. Nevertheless, these solutions are not completely feasible in practice. We suggest some ways to improve them. The second problem treated in this thesis is the shortest capacitated paths problem (SCPP). We present a Tabu Search heuristic and some post-optimization procedures for the SCPP. These procedures are called in our Tabu Search algorithm. Two versions of this algorithm are considered. In the first one, post-optimization procedures are intensively used. In the second one, they are less frequently applied. This second version is faster. Compared with a greedy approach, the two versions of our Tabu Search heuristic obtain solutions which are, on average, of better quality. The SCPP was proposed to us by EDF (Electricité de France) in order to minimize the total length of the paths followed by cables in a nuclear power-station. It can also be viewed as a subproblem of the bandwidth packing problem (BWP) appearing in telecommunications.

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Clustering is a method for discovering structure in data, widely used across many scientific disciplines. The two main clustering problems this dissertation considers are K-means and K-medoids. These are NP-hard problems in the number of samples and clusters, and both have well studied heuristic approximation algorithms. An example is Lloyd's algorithm for K-means, which is so widely used that it has become synonymous with the problem it attempts to solve.
A large part of this dissertation is about accelerating Lloyd's algorithm, and its mini-batch and K-medoids variants. The basic tool used to achieve these accelerations is the triangle inequality, which can be applied in a multitude of ways to eliminate costly distance calculations between data samples, as well as to reduce the number of comparisons of these distances.
The first effective use of the triangle inequality to accelerate K-means was by Elkan (2003) with novel refinements appearing more recently in Hamerly (2010). In Chapter 1 we extend these approaches. First, we show that by using centers stored from previous iterations, one can greatly reduce the number of sample-center distance computations, with substantial improvements in algorithm execution time. We then present an improvement over previous triangle inequality based algorithms for low-dimensions, which uses inter-center distances in a novel way.
Chapter 2 considers the use of the triangle inequality to accelerate the mini-batch variant of Lloyd's algorithm (Sculley, 2011). The main difficulty of incorporating triangle inequality bounding in this setting is that clusters can move significantly during the iterations in which a sample is unused, which makes triangle inequality bounding ineffective. We propose a modified sampling scheme to reduce the length of these periods of dormancy, and present an algorithm which achieves an order of magnitude acceleration over the standard mini-batch algorithm.
We then turn attention to the K-medoids problem. In Chapter 3 we focus on the specific problem of determining the medoid of a set. With N samples in R-d, we present a simple algorithm of complexity O(N^{3/2}) to determine the medoid. It is the first sub-quadratic algorithm for this problem when d > 1. The algorithm makes use of the triangle inequality to eliminate all but O(N^{1/2}) samples as potential medoid candidates.
Finally, in Chapter 4 we compare different K-medoids algorithms, and find that CLARANS, which iteratively replaces randomly selected centers with non-centers, avoids the local minima of other popular K-medoids algorithms. This motivates the use of CLARANS for initializing Lloyd's algorithm for K-means, which results in improved final energies as compared to K-means++ seeding. We use the triangle inequality to offset the increased computation required by CLARANS.

Background: The gene family-free framework for comparative genomics aims at providing methods for gene order analysis that do not require prior gene family assignment, but work directly on a sequence similarity graph. We study two problems related to the breakpoint median of three genomes, which asks for the construction of a fourth genome that minimizes the sum of breakpoint distances to the input genomes. Methods: We present a model for constructing a median of three genomes in this family-free setting, based on maximizing an objective function that generalizes the classical breakpoint distance by integrating sequence similarity in the score of a gene adjacency. We study its computational complexity and we describe an integer linear program (ILP) for its exact solution. We further discuss a related problem called family-free adjacencies for k genomes for the special case of k

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