Improved Approximation Algorithms for the Expanding Search Problem

A searcher faces a graph with edge lengths and vertex weights, initially having explored only a given starting vertex. In each step, the searcher adds an edge to the solution that connects an unexplored vertex to an explored vertex. This requires an amount of time equal to the edge length. The goal is to minimize the weighted sum of the exploration times over all vertices. We show that this problem is hard to approximate and provide algorithms with improved approximation guarantees. For the general case, we give a (2e+?)-approximation for any ? > 0. For the case that all vertices have unit weight, we provide a 2e-approximation. Finally, we provide a PTAS for the case of a Euclidean graph. Previously, for all cases only an 8-approximation was known

Approximation Algorithms for Capacitated k-Travelling Repairmen Problems

We study variants of the capacitated vehicle routing problem. In the multiple depot capacitated k-travelling repairmen problem (MD-CkTRP), we have a collection of clients to be served by one vehicle in a fleet of k identical vehicles based at given depots. Each client has a given demand that must be satisfied, and each vehicle can carry a total of at most Q demand before it must resupply at its original depot. We wish to route the vehicles in a way that obeys the constraints while minimizing the average time (latency) required to serve a client. This generalizes the Multi-depot k-Travelling Repairman Problem (MD-kTRP) [Chekuri and Kumar, IEEE-FOCS, 2003; Post and Swamy, ACM-SIAM SODA, 2015] to the capacitated vehicle setting, and while it has been previously studied [Lysgaard and Wohlk, EJOR, 2014; Rivera et al, Comput Optim Appl, 2015], no approximation algorithm with a proven ratio is known. We give a 42.49-approximation to this general problem, and refine this constant to 25.49 when clients have unit demands. As far as we are aware, these are the first constant-factor approximations for capacitated vehicle routing problems with a latency objective. We achieve these results by developing a framework allowing us to solve a wider range of latency problems, and crafting various orienteering-style oracles for use in this framework. We also show a simple LP rounding algorithm has a better approximation ratio for the maximum coverage problem with groups (MCG), first studied by Chekuri and Kumar [APPROX, 2004], and use it as a subroutine in our framework. Our approximation ratio for MD-CkTRP when restricted to uncapacitated setting matches the best known bound for it [Post and Swamy, ACM-SIAM SODA, 2015]. With our framework, any improvements to our oracles or our MCG approximation will result in improved approximations to the corresponding k-TRP problem

An Improved Online Algorithm for the Traveling Repairperson Problem on a Line

In the online variant of the traveling repairperson problem (TRP), requests arrive in time at points of a metric space X and must be eventually visited by a server. The server starts at a designated point of X and travels at most at unit speed. Each request has a given weight and once the server visits its position, the request is considered serviced; we call such time completion time of the request. The goal is to minimize the weighted sum of completion times of all requests. In this paper, we give a 5.429-competitive deterministic algorithm for line metrics improving over 5.829-competitive solution by Krumke et al. (TCS 2003). Our result is obtained by modifying the schedule by serving requests that are close to the origin first. To compute the competitive ratio of our approach, we use a charging scheme, and later evaluate its properties using a factor-revealing linear program which upper-bounds the competitive ratio

Improving a State-of-the-Art Heuristic for the Minimum Latency Problem with Data Mining

Recently, hybrid metaheuristics have become a trend in operations research. A successful example combines the Greedy Randomized Adaptive Search Procedures (GRASP) and data mining techniques, where frequent patterns found in high-quality solutions can lead to an efficient exploration of the search space, along with a significant reduction of computational time. In this work, a GRASP-based state-of-the-art heuristic for the Minimum Latency Problem (MLP) is improved by means of data mining techniques for two MLP variants. Computational experiments showed that the approaches with data mining were able to match or improve the solution quality for a large number of instances, together with a substantial reduction of running time. In addition, 88 new cost values of solutions are introduced into the literature. To support our results, tests of statistical significance, impact of using mined patterns, equal time comparisons and time-to-target plots are provided.Comment: This document is a dissertation fil

Un enfoque biobjetivo al problema del reparador

Objetivos y método de estudio: El objetivo principal de este trabajo consiste en el planteamiento y estudio de un problema biobjetivo con diversas aplicaciones potenciales. En este problema se tiene un conjunto de clientes que demanda algún producto o servicio. Se conoce el tiempo de servicio en cada cliente así como los tiempos de viaje entre cada par de clientes se busca encontrar la ruta que visite a todos los clientes partiendo de un depósito y volviendo a el, de forma tal que se minimice tanto la distancia recorrida por el vehículo en la ruta así como el tiempo de espera de los clientes. Para esto, se propone un modelo matemático biobjetivo en donde cada objetivo surge a partir de distintos puntos de vista. El punto de vista económico es el objetivo de distancia, en donde se desea minimizar la distancia total recorrida; desde el punto de vista social o de servicio al cliente, se tiene el objetivo de latencia en donde se desea minimizar el tiempo que el cliente espera para recibir el servicio. Se presenta una metodología de solución al problema, el cual es un algoritmo genético elitista que es muy utilizado en los problemas multiobjetivo gracias a los buenos resultados que ofrece. Contribuciones y conclusiones: La contribución de este trabajo se centra en el estudio realizado sobre un problema biobjetivo novedoso, dado que la revisión de la literatura mostró que no existen trabajos donde los objetivos que se buscó optimizar hayan sido tratados conjuntamente. Es por esto, que tanto el modelo biobjetivo presentado as´ı como el algoritmo propuesto se consideran contribuciones directas de este trabajo

TSP and its variants : use of solvable cases in heuristics

This thesis proposes heuristics motivated by solvable cases for the travelling salesman problem (TSP) and the cumulative travelling salesman path problem (CTSPP). The solvable cases are investigated in three aspects: specially structured matrices, special neighbourhoods and small-size problems. This thesis demonstrates how to use solvable cases in heuristics for the TSP and the CTSPP and presents their promising performance in theoretical research and empirical research. Firstly, we prove that the three classical heuristics, nearest neighbour, double-ended nearest neighbour and GREEDY, have the theoretical property of obtaining the permutation for permuted strong anti-Robinson matrices for the TSP such that the renumbered matrices satisfy the anti-Robinson conditions. Inspired by specially structured matrices, we propose Kalmanson heuristics, which not only have the theoretical property of solving permuted strong Kalmanson matrices to optimality for the TSP, but also outperform their classical counterparts for general cases. Secondly, we propose three heuristics for the CTSPP. The pyramidal heuristic is motivated by the special pyramidal neighbourhood. The chains heuristic and the sliding window heuristic are motivated by solvable small-size problems. The experiments suggest the proposed heuristics outperform the classical GRASP-2-opt on general cases for the CTSPP. Thirdly, we conduct both theoretical and empirical research on specially structured cases for the CTSPP. Theoretically, we prove the solvability of Line- CTSPP on more general cases and the time complexity of the CTSPP on SUM matrices. We also conjecture that the CTSPP on two rays is NP-hard. Empirically, we propose three heuristics, which perform well on specially structured cases. The Line heuristic, based on Line-CTSPP, performs better than GRASP-2-opt when nodes are distributed on two close parallel lines. The Up-Down heuristic is inspired by the Up-Down structure in solvable Path TSP and outperforms GRASP-2-opt in convex-hull cases and close-to-convex-hull cases. The Two-Ray heuristic combines the path structures in the first two heuristics and obtains high-quality solutions when nodes are along two rays

Le problème du postier chinois cumulatif

Résumé Le sujet de cette thèse est le problème du postier chinois cumulatif (PPCC). Dans ce problème, nous considérons l'importance du moment où une arête est traitée complètement. Cette façon de procéder introduit un caractère cumulatif et dynamique dans le coût réel des arêtes, ce qui a pour effet de changer la structure du problème du postier chinois. Nous démontrons que ce problème est fortement NP-difficile et réductible à une version du problème de voyageur de commerce cumulatif. Ce problème est, à notre connaissance, nouveau. Nous continuons ici l'étude entreprise dans notre mémoire de maîtrise. Notre but dans cette thèse est de résoudre exactement ce problème à l'aide des outils de la programmation linéaire en nombres entiers. Notre contribution est de plusieurs ordres. Premièrement, nous développons une vingtaine de modèles différents. Dans cette thèse, nous étudions les huit meilleurs et les comparons aussi bien empiriquement que théoriquement entre eux et démontrons toutes les relations de dominance entre eux. L'aboutissement de nos recherches est le modèle L8. Deuxièmement, nous résolvons ce modèle L8 à l'aide d'un algorithme de séparation et évaluation progressive (Branch and Cut - algorithme BC1). Nous développons plusieurs outils dont nous présentons ici trois branchements, sept pré-traitements, six familles de coupes dont trois que nous généralisons. Ces outils nous permettent déjà de battre le solveur CPLEX par un facteur de 3 à 58 sur nos graphes de référence. Troisièmement, nous développons une meilleure variante du modèle L8 : le modèle L8+ et utilisons une approche avec génération de colonnes (Branch, Price and Cut – algorithme BPC1). Dans la foulée, nous développons cinq familles de coupes et nous généralisons quatre d'entre-elles. Cette nouvelle approche, plus rapide que la première d'un facteur de 2 à 4, nous permet d'être de 2 à 133 fois plus rapide que le solveur CPLEX en utilisant le modèle L8+ sur nos graphes de référence. Quatrièmement, nous améliorons notre approche de génération de colonnes (Branch, Price and Cut – algorithme BPC2) avec une évaluation implicite du dual. Les plus grandes instances du PPCC que nous arrivons à résoudre dans un délai maximal d'une heure comprennent des graphes de 11 sommets et/ou de 55 arêtes, ce qui correspond approximativement à des instances du problème du voyageur de commerce cumulatif à 110 sommets. Mots clefs Tournées sur les arcs, fonction cumulative, problème du postier chinois cumulatif.----------Abstract The subject of this Ph.D. thesis is the Cumulative Chinese Postman Problem (CCPP). We focus on the delay of the service of each arc. This introduces a cumulative and dynamic aspect in the objective function therefore changing the structure of the Chinese Postman Problem. We prove that this problem is strongly NP-hard and reducible to a version of the Cumulative Travelling Salesman Problem. This problem is, to our knowledge, entirely new. The study of this problem was initiated in our master thesis. Our main goal in this thesis is to solve this problem exactly with the help of the tools of linear integer programming. Our contribution is manifold. First, we develop twenty different models. However, in this thesis, we only discuss and compare theoretically and experimentally the best eight models. We prove all dominance relations among them. Model L8 stands out as the best model. Secondly, we solve this model L8 with a Branch and Cut (algorithm BC1). Throughout our study, we develop several tools among which three branching rules, seven presolving algorithms, six families of cuts (three of them generalized). These tools alone allow us to solve the problem faster than CPLEX by a factor of 3 to 58 on our test graphs. Thirdly, we develop an improved model L8+ and use a column generation approach - a Branch, Price and Cut (algorithm BPC1). We also develop five new families of cuts (four of them generalized). This new approach is faster than the previous one by a factor of 2 to 4 and is faster than CPLEX with the new model L8+ by a factor of 2 to 133 on our test graphs. Fourthly, we improve our Branch, Price and Cut algorithm (algorithm BPC2) by using an implicit evaluation of the dual. The largest instances for which we are able to solve the CCPP in less than one hour include graphs with 11 nodes and/or 55 edges which correspond approximately to instances of the Cumulative Travelling Salesman Problem with 110 nodes