@misc{AscheuerKrumkeRambau, author = {Ascheuer, Norbert and Krumke, Sven and Rambau, J{\"o}rg}, title = {The Online Transportation Problem: Competitive Scheduling of Elevators}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3779}, number = {SC-98-34}, abstract = {In this paper we consider the following online transportation problem (\textsc{Oltp}): Objects are to be transported between the vertices of a given graph. Transportation requests arrive online, specifying the objects to be transported and the corresponding source and target vertex. These requests are to be handled by a server which commences its work at a designated origin vertex and which picks up and drops objects at their starts and destinations. After the end of its service the server returns to its start. The goal of \textsc{Oltp} is to come up with a transportation schedule for the server which finishes as early as possible. We first show a lower bound of~\$5/3\$ for the competitive ratio of any deterministic algorithm. We then analyze two simple and natural strategies which we call \textsf{REPLAN} and \textsf{IGNORE}. \textsf{REPLAN} completely discards its schedule and recomputes a new one when a new request arrives. \textsf{IGNORE} always runs a (locally optimal) schedule for a set of known requests and ignores all new requests until this schedule is completed. We show that both strategies, \textsf{REPLAN} and \textsf{IGNORE}, are \$5/2\$-competitive. We also present a somewhat less natural strategy \textsf{SLEEP}, which in contrast to the other two strategies may leave the server idle from time to time although unserved requests are known. We also establish a competitive ratio of~\$5/2\$ for the algorithm \textsf{SLEEP}. Our results are extended to the case of ``open schedules'' where the server is not required to return to its start position at the end of its service.}, language = {en} } @misc{BlomKrumkePaepeetal., author = {Blom, Michiel and Krumke, Sven and Paepe, Willem de and Stougie, Leen}, title = {The Online-TSP Against Fair Adversaries}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-5779}, number = {00-09}, abstract = {In the online traveling salesman problem requests for visits to cities (points in a metric space) arrive online while the salesman is traveling. The salesman moves at no more than unit speed and starts and ends his work at a designated origin. The objective is to find a routing for the salesman which finishes as early as possible. Performance of algorithms is measured through their competitive ratio, comparing the outcome of the algorithms with that of an adversary who provides the problem instance and therefore is able to achieve the optimal offline solution. Objections against such omnipotent adversaries have lead us to devise an adversary that is in a natural way, in the context of routing problems, more restricted in power. For the exposition we consider the online traveling salesman problem on the metric space given by the non-negative part of the real line. We show that a very natural strategy is~\$3/2\$-competitive against the conventional adversary, which matches the lower bound on competitive ratios achievable for algorithms for this problem. Against the more ``\emph{fair adversary}'', that we propose, we show that there exists an algorithm with competitive ratio \$\frac{1+\sqrt{17}}{4}\approx 1.28\$ and provide a matching lower bound. We also show competitiveness results for a special class of algorithms (called zealous algorithms) that do not allow waiting time for the server as long as there are requests unserved.}, language = {en} } @misc{KrumkeRambau, author = {Krumke, Sven and Rambau, J{\"o}rg}, title = {Probieren geht {\"u}ber Studieren? Entscheidungshilfen f{\"u}r kombinatorische Online-Optimierungsprobleme in der innerbetrieblichen Logistik}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6723}, number = {02-05}, abstract = {Die Automatisierung von innerbetrieblicher Logistik erfordert -- {\"u}ber die physikalische Steuerung von Ger{\"a}ten hinaus -- auch eine effiziente Organisation der Transporte: ein Aufgabenfeld der kombinatorischen Optimierung. Dieser Artikel illustriert anhand von konkreten Aufgabenstellungen die Online-Problematik (unvollst{\"a}ndiges Wissen) sowie die Echtzeit-Problematik (beschr{\"a}nkte Rechenzeit), auf die man in der innerbetrieblichen Logistik trifft. Der Text gibt einen {\"U}berblick {\"u}ber allgemeine Konstruktionsprinzipien f{\"u}r Online-Algorithmen und Bewertungsmethoden, die bei der Entscheidung helfen, welche Algorithmen f{\"u}r eine vorliegende Problemstellung geeignet sind.}, language = {de} } @misc{Krumke, author = {Krumke, Sven}, title = {News from the Online Traveling Repairman}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-5767}, number = {00-08}, abstract = {The traveling repairman problem (TRP) is a variant of the famous traveling salesman problem (TSP). The objective for the TRP is to minimize the latency, that is the the weighted sum of completion times of the cities, where the completion time of a city is defined to be the time in the tour before the city is reached. In the online traveling repairman problem (OLTRP) requests for visits to cities (points in a metric space) arrive online while the repairman is traveling. We analyze the performance of algorithms using competitive analysis, where the cost of an online algorithm is compared to that of an optimal offline algorithm. An optimal offline algorithm knows the entire request sequence in advance and can serve it with minimum cost. Recently, Feuerstein and Stougie presented a \$9\$-competitive algorithm for the OLTRP on the real line. In this paper we show how to use techniques from online-scheduling to obtain an \$8\$-competitive deterministic algorithm which works for any metric space. We also present a randomized algorithm which has a competitive ratio of \$\frac{4}{\ln 2}\approx 5.7708\$ against an oblivious adversary. All of our results also hold for the ``dial-a-ride'' generalization of the OLTRP, where objects have to be picked up and delivered by a server.}, language = {en} } @misc{HauptmeierKrumkeRambau, author = {Hauptmeier, Dietrich and Krumke, Sven and Rambau, J{\"o}rg}, title = {The Online Dial-a-Ride Problem under Reasonable Load}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3961}, number = {SC-99-08}, abstract = {In this paper, we analyze algorithms for the online dial-a-ride problem with request sets that fulfill a certain worst-case restriction: roughly speaking, a set of requests for the online dial-a-ride problem is reasonable if the requests that come up in a sufficiently large time period can be served in a time period of at most the same length. This new notion is a stability criterion implying that the system is not overloaded. The new concept is used to analyze the online dial-a-ride problem for the minimization of the maximal resp.\ average flow time. Under reasonable load it is possible to distinguish the performance of two particular algorithms for this problem, which seems to be impossible by means of classical competitive analysis.}, language = {en} } @misc{GroetschelHauptmeierKrumkeetal., author = {Gr{\"o}tschel, Martin and Hauptmeier, Dietrich and Krumke, Sven and Rambau, J{\"o}rg}, title = {Simulation Studies for the Online-Dial-a-Ride Problem}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3976}, number = {SC-99-09}, abstract = {In a large distribution center of Herlitz AG, Berlin, we invesigated the elevator subsystem of the fully automated pallet transportation system. Each elevator may carry one pallet and has to serve eight levels. The goal is to minimize the average resp.\ the maximum flow time. The variants of this elevator control problem have been subject of recent theoretical research and are known as online-dial-a-ride problems. In this paper we investigate several online algorithms for several versions of online-dial-a-ride problems by means of a simulation program, developed on the basis of the simulation library AMSEL. We draw statistics from samples of randomly generated data providing for different load situations. Moreover, we provide preliminary studies with real production data for a system of five elevators connected by a conveyor circuit, as can be found at the Herlitz plant. We show which algorithms are best under certain load situations and which lead to break downs under particular circumstances.}, language = {en} } @misc{BorndoerferDaneckerWeiser, author = {Bornd{\"o}rfer, Ralf and Danecker, Fabian and Weiser, Martin}, title = {A Discrete-Continuous Algorithm for Free Flight Planning}, issn = {1438-0064}, doi = {10.3390/a14010004}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-81343}, abstract = {We propose a hybrid discrete-continuous algorithm for flight planning in free flight airspaces. In a first step, our DisCOptER method discrete-continuous optimization for enhanced resolution) computes a globally optimal approximate flight path on a discretization of the problem using the A* method. This route initializes a Newton method that converges rapidly to the smooth optimum in a second step. The correctness, accuracy, and complexity of the method are goverened by the choice of the crossover point that determines the coarseness of the discretization. We analyze the optimal choice of the crossover point and demonstrate the asymtotic superority of DisCOptER over a purely discrete approach.}, language = {en} } @article{BorndoerferDaneckerWeiser2020, author = {Bornd{\"o}rfer, Ralf and Danecker, Fabian and Weiser, Martin}, title = {A Discrete-Continuous Algorithm for Free Flight Planning}, series = {Algorithms}, volume = {14}, journal = {Algorithms}, number = {1}, publisher = {MDPI}, issn = {1438-0064}, doi = {10.3390/a14010004}, pages = {4}, year = {2020}, abstract = {We propose a hybrid discrete-continuous algorithm for flight planning in free flight airspaces. In a first step, our DisCOptER method discrete-continuous optimization for enhanced resolution) computes a globally optimal approximate flight path on a discretization of the problem using the A* method. This route initializes a Newton method that converges rapidly to the smooth optimum in a second step. The correctness, accuracy, and complexity of the method are goverened by the choice of the crossover point that determines the coarseness of the discretization. We analyze the optimal choice of the crossover point and demonstrate the asymtotic superority of DisCOptER over a purely discrete approach.}, language = {en} } @misc{BortolettoLindnerMasing, author = {Bortoletto, Enrico and Lindner, Niels and Masing, Berenike}, title = {The tropical and zonotopal geometry of periodic timetables}, issn = {1438-0064}, abstract = {The Periodic Event Scheduling Problem (PESP) is the standard mathematical tool for optimizing periodic timetabling problems in public transport. A solution to PESP consists of three parts: a periodic timetable, a periodic tension, and integer periodic offset values. While the space of periodic tension has received much attention in the past, we explore geometric properties of the other two components, establishing novel connections between periodic timetabling and discrete geometry. Firstly, we study the space of feasible periodic timetables, and decompose it into polytropes, i.e., polytopes that are convex both classically and in the sense of tropical geometry. We then study this decomposition and use it to outline a new heuristic for PESP, based on the tropical neighbourhood of the polytropes. Secondly, we recognize that the space of fractional cycle offsets is in fact a zonotope. We relate its zonotopal tilings back to the hyperrectangle of fractional periodic tensions and to the tropical neighbourhood of the periodic timetable space. To conclude we also use this new understanding to give tight lower bounds on the minimum width of an integral cycle basis.}, language = {en} }