Online k-server routing problems

In an online k-server routing problem, a crew of k servers has to visit points in a metric space as they arrive in real time. Possible objective functions include minimizing the makespan (k-Traveling Salesman Problem) and minimizing the sum of completion times (k-Traveling Repairman Problem). We give competitive algorithms, resource augmentation results and lower bounds for k-server routing problems in a wide class of metric spaces. In some cases the competitive ratio is dramatically better than that of the corresponding single server problem. Namely, we give a 1+O((log¿k)/k)-competitive algorithm for the k-Traveling Salesman Problem and the k-Traveling Repairman Problem when the underlying metric space is the real line. We also prove that a similar result cannot hold for the Euclidean plane

Derandomization of Online Assignment Algorithms for Dynamic Graphs

This paper analyzes different online algorithms for the problem of assigning weights to edges in a fully-connected bipartite graph that minimizes the overall cost while satisfying constraints. Edges in this graph may disappear and reappear over time. Performance of these algorithms is measured using simulations. This paper also attempts to derandomize the randomized online algorithm for this problem

Online Assignment Algorithms for Dynamic Bipartite Graphs

This paper analyzes the problem of assigning weights to edges incrementally in a dynamic complete bipartite graph consisting of producer and consumer nodes. The objective is to minimize the overall cost while satisfying certain constraints. The cost and constraints are functions of attributes of the edges, nodes and online service requests. Novelty of this work is that it models real-time distributed resource allocation using an approach to solve this theoretical problem. This paper studies variants of this assignment problem where the edges, producers and consumers can disappear and reappear or their attributes can change over time. Primal-Dual algorithms are used for solving these problems and their competitive ratios are evaluated

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

Exploration of Graphs with Excluded Minors

We study the online graph exploration problem proposed by Kalyanasundaram and Pruhs (1994) and prove a constant competitive ratio on minor-free graphs. This result encompasses and significantly extends the graph classes that were previously known to admit a constant competitive ratio. The main ingredient of our proof is that we find a connection between the performance of the particular exploration algorithm Blocking and the existence of light spanners. Conversely, we exploit this connection to construct light spanners of bounded genus graphs. In particular, we achieve a lightness that improves on the best known upper bound for genus g ? 1 and recovers the known tight bound for the planar case (g = 0)

Exploration of graphs with excluded minors

We study the online graph exploration problem proposed by Kalyanasundaram and Pruhs (1994) and prove a constant competitive ratio on minor-free graphs. This result encompasses and significantly extends the graph classes that were previously known to admit a constant competitive ratio. The main ingredient of our proof is that we find a connection between the performance of the particular exploration algorithm Blocking and the existence of light spanners. Conversely, we exploit this connection to construct light spanners of bounded genus graphs. In particular, we achieve a lightness that improves on the best known upper bound for genus g>0 and recovers the known tight bound for the planar case (g=0).Comment: to appear at ESA 202

Traveling Repairperson, Unrelated Machines, and Other Stories About Average Completion Times

We present a unified framework for minimizing average completion time for many seemingly disparate online scheduling problems, such as the traveling repairperson problems (TRP), dial-a-ride problems (DARP), and scheduling on unrelated machines. We construct a simple algorithm that handles all these scheduling problems, by computing and later executing auxiliary schedules, each optimizing a certain function on already seen prefix of the input. The optimized function resembles a prize-collecting variant of the original scheduling problem. By a careful analysis of the interplay between these auxiliary schedules, and later employing the resulting inequalities in a factor-revealing linear program, we obtain improved bounds on the competitive ratio for all these scheduling problems. In particular, our techniques yield a 4-competitive deterministic algorithm for all previously studied variants of online TRP and DARP, and a 3-competitive one for the scheduling on unrelated machines (also with precedence constraints). This improves over currently best ratios for these problems that are 5.14 and 4, respectively. We also show how to use randomization to further reduce the competitive ratios to 1+2/ln 3 < 2.821 and 1+1/ln 2 < 2.443, respectively. The randomized bounds also substantially improve the current state of the art. Our upper bound for DARP contradicts the lower bound of 3 given by Fink et al. (Inf. Process. Lett. 2009); we pinpoint a flaw in their proof

Dynamic vehicle routing problems: Three decades and counting

Since the late 70s, much research activity has taken place on the class of dynamic vehicle routing problems (DVRP), with the time period after year 2000 witnessing a real explosion in related papers. Our paper sheds more light into work in this area over more than 3 decades by developing a taxonomy of DVRP papers according to 11 criteria. These are (1) type of problem, (2) logistical context, (3) transportation mode, (4) objective function, (5) fleet size, (6) time constraints, (7) vehicle capacity constraints, (8) the ability to reject customers, (9) the nature of the dynamic element, (10) the nature of the stochasticity (if any), and (11) the solution method. We comment on technological vis-à-vis methodological advances for this class of problems and suggest directions for further research. The latter include alternative objective functions, vehicle speed as decision variable, more explicit linkages of methodology to technological advances and analysis of worst case or average case performance of heuristics.© 2015 Wiley Periodicals, Inc

The traveling repairman problem

Diese Magisterarbeit gibt einen Überblick über das Traveling Repairman Problem (TRP), das eine Spezialform des Problems des Handlungsreisenden (Traveling Salesman Problem – TSP) darstellt. Beide Modelle werden benutzt, um die Tour eines Handlungsreisenden zu planen, der in einer vorgegebenen Zeitspanne eine bestimmte Anzahl von Kunden besuchen soll. Während das TSP sich darauf konzentriert, die Länge der Tour zu minimieren, versucht das TRP, die Summe der Wartezeiten der Kunden so gering wie möglich zu halten. Der Hauptteil der Arbeit beschäftigt sich mit der Definition und den Varianten des TRP und beschreibt mögliche Modelle und Verfahren, mit deren Hilfe diese zu lösen sind. Dabei werden zuerst die Problemstellungen definiert und dann die mathematischen Formulierungen bzw. die Algorithmen dargestellt. Zu Beginn der Arbeit werden das TSP und das TRP näher definiert und kurz anhand eines Beispiels illustriert (in Kapitel 2). Danach werden das allgemeine TRP und einige Lösungsverfahren dazu näher erläutert (in Kapitel 3). Im Hauptteil werden zuerst einige Variationen des TRP mit einem einzelnen Repairman und Algorithmen zur Lösung dieser Modelle beschrieben (in Kapitel 4). Dann werden das TRP mit mehreren Repairmen sowie einige Spezialformen hierzu erläutert (in Kapitel 5). Zusätzlich werden in dieser Arbeit Anwendungsmöglichkeiten beschrieben, von denen zwei genauer untersucht werden (in Kapitel 6). Schließlich werden noch einige Basisbegriffe und Lösungsmethoden erläutert (in Kapitel 7)