Improving Christofides' Algorithm for the s-t Path TSP

We present a deterministic (1+sqrt(5))/2-approximation algorithm for the s-t path TSP for an arbitrary metric. Given a symmetric metric cost on n vertices including two prespecified endpoints, the problem is to find a shortest Hamiltonian path between the two endpoints; Hoogeveen showed that the natural variant of Christofides' algorithm is a 5/3-approximation algorithm for this problem, and this asymptotically tight bound in fact has been the best approximation ratio known until now. We modify this algorithm so that it chooses the initial spanning tree based on an optimal solution to the Held-Karp relaxation rather than a minimum spanning tree; we prove this simple but crucial modification leads to an improved approximation ratio, surpassing the 20-year-old barrier set by the natural Christofides' algorithm variant. Our algorithm also proves an upper bound of (1+sqrt(5))/2 on the integrality gap of the path-variant Held-Karp relaxation. The techniques devised in this paper can be applied to other optimization problems as well: these applications include improved approximation algorithms and improved LP integrality gap upper bounds for the prize-collecting s-t path problem and the unit-weight graphical metric s-t path TSP.Comment: 31 pages, 5 figure

FIXED RATIO POLYNOMIAL TIME APPROXIMATION ALGORITHM FOR THE PRIZE-COLLECTING ASYMMETRIC TRAVELING SALESMAN PROBLEM

We develop the first fixed-ratio approximation algorithm for the well-known Prize-Collecting Asymmetric Traveling Salesman Problem,  which has numerous valuable applications in operations research. An instance of this problem is given by a complete node- and edge-weighted digraph $G$. Each node of the graph $G$ can either be visited by the resulting route or skipped, for some penalty, while the arcs of $G$  are weighted by non-negative transportation costs that fulfill the triangle inequality constraint. The goal is to find a closed walk that minimizes the total transportation costs augmented by the accumulated penalties. We show that an arbitrary $\alpha$-approximation algorithm for the Asymmetric Traveling Salesman Problem induces an $(\alpha+1)$-approximation for the problem in question. In particular, using the recent $(22+\varepsilon)$-approximation algorithm of V. Traub and J. Vygen that improves the seminal result of  O. Svensson, J. Tarnavski, and L. Végh,  we obtain $(23+\varepsilon)$-approximate solutions for the problem

Underwater Data Collection Using Robotic Sensor Networks

We examine the problem of utilizing an autonomous underwater vehicle (AUV) to collect data from an underwater sensor network. The sensors in the network are equipped with acoustic modems that provide noisy, range-limited communication. The AUV must plan a path that maximizes the information collected while minimizing travel time or fuel expenditure. We propose AUV path planning methods that extend algorithms for variants of the Traveling Salesperson Problem (TSP). While executing a path, the AUV can improve performance by communicating with multiple nodes in the network at once. Such multi-node communication requires a scheduling protocol that is robust to channel variations and interference. To this end, we examine two multiple access protocols for the underwater data collection scenario, one based on deterministic access and another based on random access. We compare the proposed algorithms to baseline strategies through simulated experiments that utilize models derived from experimental test data. Our results demonstrate that properly designed communication models and scheduling protocols are essential for choosing the appropriate path planning algorithms for data collection.United States. Office of Naval Research (ONR N00014-09-1-0700)United States. Office of Naval Research (ONR N00014-07-1-00738)National Science Foundation (U.S.) (NSF 0831728)National Science Foundation (U.S.) (NSF CCR-0120778)National Science Foundation (U.S.) (NSF CNS-1035866

A Swarm of Salesmen: Algorithmic Approaches to Multiagent Modeling

This honors thesis describes the algorithmic abstraction of a problem modeling a swarm of Mars rovers, where many agents must together achieve a goal. The algorithmic formulation of this problem is based on the traveling salesman problem (TSP), and so in this thesis I offer a review of the mathematical technique of linear programming in the context of its application to the TSP, an overview of some variations of the TSP and algorithms for approximating and solving them, and formulations without solutions of two novel TSP variations which are useful for modeling the original problem

Problema geométrico conexo de localização de instalações

Orientadores: Flávio Keidi Miyazawa, Rafael Crivellari Saliba SchoueryDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: Esse trabalho visa estudar problemas da família Localização de Instalações. Nesses problemas, recebemos de entrada um conjunto de clientes e um conjunto de instalações. Queremos encontrar e abrir um subconjunto de instalações, normalmente, pagando um preço por cada instalação aberta. Nosso objetivo é conectar clientes a instalações abertas, pagando o menor custo possível. Esse problema tem grandes aplicações na área de Pesquisa Operacional e Telecomunicações. Estamos especialmente interessados no problema de Localização de Instalações Geométrico e Conexo. Nessa versão do problema, as instalações podem ser abertas em qualquer lugar de um plano de dimensão d, e pagamos um preço fixo f por cada instalação aberta. Também devemos conectar as instalações entre si formando uma árvore. Essa árvore normalmente recebe uma ponderação maior, uma vez que suas conexões agregam atendimento para quantidade maior de recursos. Para representar tal ponderação seus custos são multiplicados por um parametro M > 0 dado como parte da entrada. Apresentamos um Esquema de Aproximação Polinomial para a versão euclidiana do problemaAbstract: In this work we study problems from the facility location family. In this set of problems, we want to find and open a subset of given facilities. Usually, a price is paid for each opened facility. Our goal is to connect given clients to the closest opened facilities incurring in the smallest cost possible. This problem has several practical applications in Operations Research and Telecommunication. We are specially interested in the Geometric Connected Facility Location problem. In this version, facilities can be anywhere in the d-dimensional plane, and we have to pay a fixed price f to open each facility. We also have the requirement of connecting facilities among themselves forming a tree. This tree is usually weighted by a given parameter M > 0. We present a Polynomial-Time Approximation Scheme for the two-dimensional version of this problemMestradoCiência da ComputaçãoMestra em Ciência da Computação1406904, 1364124CAPE

A Swarm of Salesmen: Algorithmic Approaches to Multiagent Modeling

This honors thesis describes the algorithmic abstraction of a problem modeling a swarm of Mars rovers, where many agents must together achieve a goal. The algorithmic formulation of this problem is based on the traveling salesman problem (TSP), and so in this thesis I offer a review of the mathematical technique of linear programming in the context of its application to the TSP, an overview of some variations of the TSP and algorithms for approximating and solving them, and formulations without solutions of two novel TSP variations which are useful for modeling the original problem

Algoritmos de aproximação para problemas de roteamento e conectividade com múltiplas funções de distância

Orientador: Lehilton Lelis Chaves PedrosaDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: Nesta dissertação, estudamos algumas generalizações de problemas clássicos de roteamento e conectividade cujas instâncias são compostas por um grafo completo e múltiplas funções de distância. Por exemplo, existe o Problema do Caixeiro Alugador (CaRS), no qual um viajante deseja visitar um conjunto de cidades alugando um ou mais carros disponíveis. Cada carro tem uma função de distância e uma taxa de retorno ao local do aluguel. CaRS é uma generalização do Problema do Caixeiro Viajante (TSP). Nós lidamos com esses problemas usando algoritmos de aproximação, que são algoritmos eficientes que produzem soluções com garantia de qualidade. Neste trabalho, são apresentadas duas abordagens, uma baseada em uma redução linear que preserva o fator de aproximação e outra baseada na construção de instâncias de dois problemas distintos. Os problemas considerados são o Steiner TSP, o Problema do Passeio com Coleta de Prêmios e o Problema da Floresta Restrita. Generalizamos cada um desses problemas considerando múltiplas funções de distância e, para cada um deles, apresentamos um algoritmo de aproximação com fator O(logn), onde n é o número de vértices (cidades). Essas aproximações são assintoticamente ótimas, já que não há algoritmos com fator o(log n), a não ser que P = NPAbstract: In this dissertation, we study some generalizations of classical routing and connectivity problems whose instances are composed of a complete graph and multiple distance functions. As an example, there is the Traveling Car Renter Problem (CaRS) in which a traveler wants to visit a set of cities by renting one or more available cars. Each car is associated to a distance function and a service fee to return to the rental location. CaRS is a generalization of the Traveling Salesman Problem (TSP). We deal with these problems using approximation algorithms which are efficient algorithms that produce solutions with quality guarantee. In this work, two approaches are presented, one based on a linear reduction that preserves the approximation factor and the other based on the construction of instances of two distinct problems. The studied problems are the Steiner TSP, the Profitable Tour Problem, and the Constrained Forest Problem. We generalize these problems by considering multiple distance functions and, for each of them, we present an O(log n)-approximation algorithm, where n is the number of vertices (cities). The factor is asymptotically optimal, since there is no approximation algorithm with factor o(log n) unless P = NPMestradoCiência da ComputaçãoMestra em Ciência da Computação001CAPE

A Constant-Factor Approximation for Directed Latency in Quasi-Polynomial Time

We give the first constant-factor approximation for the Directed Latency problem in quasi-polynomial time. Here, the goal is to visit all nodes in an asymmetric metric with a single vehicle starting at a depot $r$ to minimize the average time a node waits to be visited by the vehicle. The approximation guarantee is an improvement over the polynomial-time $O(\log n)$-approximation [Friggstad, Salavatipour, Svitkina, 2013] and no better quasi-polynomial time approximation algorithm was known. To obtain this, we must extend a recent result showing the integrality gap of the Asymmetric TSP-Path LP relaxation is bounded by a constant [K\"{o}hne, Traub, and Vygen, 2019], which itself builds on the breakthrough result that the integrality gap for standard Asymmetric TSP is also a constant [Svensson, Tarnawsi, and Vegh, 2018]. We show the standard Asymmetric TSP-Path integrality gap is bounded by a constant even if the cut requirements of the LP relaxation are relaxed from $x(\delta^{in}(S)) \geq 1$ to $x(\delta^{in}(S)) \geq \rho$ for some constant $1/2 < \rho \leq 1$. We also give a better approximation guarantee in the special case of Directed Latency in regret metrics where the goal is to find a path $P$ minimize the average time a node $v$ waits in excess of $c_{rv}$, i.e. $\frac{1}{|V|} \cdot \sum_{v \in V} (c_v(P)-c_{rv})$

Algorithms for Multiple Vehicle Routing Problems

Surveillance and monitoring applications require a collection of heterogeneous vehicles to visit a set of targets. This dissertation considers three fundamental routing problems involving multiple vehicles that arise in these applications. The main objective of this dissertation is to develop novel approximation algorithms for these routing problems that find feasible solutions and also provide a bound on the quality of the solutions produced by the algorithms. The first routing problem considered is a multiple depot, multiple terminal, Hamiltonian Path problem. Given multiple vehicles starting at distinct depots, a set of targets and terminal locations, the objective of this problem is to find a vertex-disjoint path for each vehicle such that each target is visited once by a vehicle, the paths end at the terminals and the sum of the distances travelled by the vehicles is a minimum. A 2-approximation algorithm is presented for this routing problem when the costs are symmetric and satisfy the triangle inequality. For the case where all the vehicles start from the same depot, a 5/3-approximation algorithm is developed. The second routing problem addressed in this dissertation is a multiple depot, heterogeneous traveling salesman problem. The objective of this problem is to find a tour for each vehicle such that each of the targets is visited at least once by a vehicle and the sum of the distances travelled by the vehicles is minimized. A primal-dual algorithm with an approximation ratio of 2 is presented for this problem when the vehicles involved are ground vehicles that can move forwards and backwards with a constraint on their minimum turning radius. Finally, this dissertation addresses a multiple depot heterogeneous traveling salesman problem when the travel costs are asymmetric and satisfy the triangle inequality. An approximation algorithm and a heuristic is developed for this problem with simulation results that corroborate the performance of the proposed algorithms. All the main algorithms presented in the dissertation advance the state of art in the area of approximation algorithms for multiple vehicle routing problems. This dissertation has its value for providing approximation algorithms for the routing problems that involves multiple vehicles with additional constraints. Some algorithms have constant approximation factor, which is very useful in the application but difficult to find. In addition to the approximation algorithms, some heuristic algorithms were also proposed to improve solution qualities or computation time

Revenue Maximization in Transportation Networks

We study the joint optimization problem of pricing trips in a transportation network and serving the induced demands by routing a fleet of available service vehicles to maximize revenue. Our framework encompasses applications that include traditional transportation networks (e.g., airplanes, buses) and their more modern counterparts (e.g., ride-sharing systems). We describe a simple combinatorial model, in which each edge in the network is endowed with a curve that gives the demand for traveling between its endpoints at any given price. We are supplied with a number of vehicles and a time budget to serve the demands induced by the prices that we set, seeking to maximize revenue. We first focus on a (preliminary) special case of our model with unit distances and unit time horizon. We show that this version of the problem can be solved optimally in polynomial time. Switching to the general case of our model, we first present a two-stage approach that separately optimizes for prices and routes, achieving a logarithmic approximation to revenue in the process. Next, using the insights gathered in the first two results, we present a constant factor approximation algorithm that jointly optimizes for prices and routes for the supply vehicles. Finally, we discuss how our algorithms can handle capacitated vehicles, impatient demands, and selfish (wage-maximizing) drivers