## Randomized Online Algorithms for Dynamic Multi-Period Routing Problems

Please always quote using this URN: urn:nbn:de:0297-zib-11132
• The Dynamic Multi-Period Routing Problem DMPRP introduced by Angelelli et al. gives a model for a two-stage online-offline routing problem. At the beginning of each time period a set of customers becomes known. The customers need to be served either in the current time period or in the following. Postponed customers have to be served in the next time period. The decision whether to postpone a customer has to be done online. At the end of each time period, an optimal tour for the customers assigned to this period has to be computed and this computation can be done offline. The objective of the problem is to minimize the distance traveled over all planning periods assuming optimal routes for the customers selected in each period. We provide the first randomized online algorithms for the DMPRP which beat the known lower bounds for deterministic algorithms. For the special case of two planning periods we provide lower bounds on the competitive ratio of any randomized online algorithm against the oblivious adversary. We identify a randomized algorithm that achieves the optimal competitive ratio of $\frac{1+\sqrt{2}}{2}$ for two time periods on the real line. For three time periods, we give a randomized algorithm that is strictly better than any deterministic algorithm.

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Author: Benjamin Hiller, Sven O. Krumke, Sleman Saliba, Andreas Tuchscherer ZIB-Report Online-Optimierung; Randomisierte Algorithmen; Traveling-Salesman-Problem; Zweistufiges Problemonline optimization; randomized algorithm; traveling salesman problem; two-stage problem 68-XX COMPUTER SCIENCE (For papers involving machine computations and programs in a specific mathematical area, see Section -04 in that area) / 68Wxx Algorithms (For numerical algorithms, see 65-XX; for combinatorics and graph theory, see 05C85, 68Rxx) / 68W40 Analysis of algorithms [See also 68Q25] 2009/02/11 ZIB-Report (09-03) 1438-0064 Appeared in: 9th Workshop on Models and Algorithms for Planning and Scheduling Problems (C. Stein, M. Uetz und T. Vredeveld, Hg.), 2009