**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

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

EPFL thesis

Abstract

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.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related concepts

Loading

Related publications

Loading

Related concepts (20)

Related publications (41)

Algorithm

In mathematics and computer science, an algorithm (ˈælɡərɪðəm) is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algo

Shortest path problem

In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized.
The pro

Heuristic

A heuristic (hjʊˈrɪstɪk; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, b

Loading

Loading

Loading

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

2017Clustering 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.

The common point between the different chapters of the present work is graph theory. We investigate some well known graph theory problems, and some which arise from more specific applications. In the first chapter, we deal with the maximum stable set problem, and provide some new graph classes, where it can be solved in polynomial time. Those classes are hereditary, i.e. characterized by a list of forbidden induced subgraphs. The algorithms proposed are purely combinatorial. The second chapter is devoted to the study of a problem linked to security purposes in mobile telecommunication networks. The particularity is that there is no central authority guaranteeing security, but it is actually managed by the users themselves. The network is modelled by an oriented graph, whose vertices represent the users, and whose arcs represent public key certificates. The problem is to associate to each vertex a subgraph with some requirements on the size of the subgraphs, the number of times a vertex is taken in a subgraph and the connectivity between any two users as they put their subgraphs together. Constructive heuristics are proposed, bounds on the optimal solution and a tabu search are described and tested. The third chapter is on the problem of reconstructing an image, given its projections in terms of the number of occurrences of each color in each row and each column. The case of two colors is known to be polynomially solvable, it is NP-complete with four or more colors, and the complexity status of the problem with three colors is open. An intermediate case between two and three colors is shown to be solvable in polynomial time. The last two chapters are about graph (vertex-)coloring. In the fourth, we prove a result which brings a large collection of NP-hard subcases, characterized by forbidden induced subgraphs. In the fifth chapter, we approach the problem with the use of linear programming. Links between different formulations are pointed out, and some families of facets are characterized. In the last section, we study a branch and bound algorithm, whose lower bounds are given by the optimal value of the linear relaxation of one of the exposed formulations. A preprocessing procedure is proposed and tested.