Abstract
We consider the online Dynamic Traveling Repair Problem (DTRP) with an arbitrary size time window. In this problem we receive a sequence of requests for service at nodes in a metric space and a time window for each request. The goal is to maximize the number of requests served during their time window. The time to traverse between two points is equal to the distance. Serving a request requires unit time. Irani et al., SODA 2002 considered the special case of a fixed size time window. In contrast, we consider the general case of an arbitrary size time window. We characterize the competitive ratio for each metric space separately. The competitive ratio depends on the relation between the minimum laxity (the minimum length of a time window) and the diameter of the metric space. Specifically, there exists a constant competitive algorithm only when the laxity is larger than the diameter. In addition, we characterize the rate of convergence of the competitive ratio, which approaches 1, as the laxity increases. Specifically, we provide matching lower and upper bounds. These bounds depend on the ratio between the laxity and the optimal TSP solution of the metric space (the minimum distance to traverse all nodes). An application of our result improves the previously known lower bound for colored packets with transition costs and matches the known upper bound. In proving our lower bounds we use an embedding with some special properties.
Supported in part by the Israel Science Foundation (grant No. 1506/16), by the Israeli Centers of Research Excellence (I-CORE) program, (Center No. 4/11) and by the Blavatnik Fund.
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References
Arkin, E.M., Mitchell, J.S., Narasimhan, G.: Resource-constrained geometric network optimization. In: SOCG, pp. 307–316. ACM (1998)
Ausiello, G., Feuerstein, E., Leonardi, S., Stougie, L., Talamo, M.: Algorithms for the on-line travelling salesman. Eindhoven University of Technology, Department of Mathematics and Computing Sciences (1999)
Awerbuch, B., Azar, Y., Blum, A., Vempala, S.: New approximation guarantees for minimum-weight k-trees and prize-collecting salesmen. SIAM J. Comput. 28(1), 254–262 (1998)
Azar, Y., Feige, U., Gamzu, I., Moscibroda, T., Raghavendra, P.: Buffer management for colored packets with deadlines. In: SPAA 2009, pp. 319–327. ACM (2009)
Bansal, N., Blum, A., Chawla, S., Meyerson, A.: Approximation algorithms for Deadline-TSP and vehicle routing with time-windows. In: STOC, pp. 166–174. ACM (2004)
Bar-Yehuda, R., Even, G., Shahar, S.M.: On approximating a geometric prize-collecting traveling salesman problem with time windows. J. Algorithms 55(1), 76–92 (2005)
Blum, A., Chawla, S., Karger, D.R., Lane, T., Meyerson, A., Minkoff, M.: Approximation algorithms for orienteering and discounted-reward TSP. In: Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, pp. 46–55. IEEE (2003)
Chekuri, C., Korula, N.: Approximation algorithms for orienteering with time windows. arXiv preprint arXiv:0711.4825 (2007)
Chekuri, C., Korula, N., Pál, M.: Improved algorithms for orienteering and related problems. ACM Trans. Algorithms (TALG) 8(3), 23 (2012)
Chekuri, C., Kumar, A.: Maximum coverage problem with group budget constraints and applications. In: Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds.) APPROX/RANDOM -2004. LNCS, vol. 3122, pp. 72–83. Springer, Heidelberg (2004). doi:10.1007/978-3-540-27821-4_7
Desrochers, M., Desrosiers, J., Solomon, M.: A new optimization algorithm for the vehicle routing problem with time windows. Oper. Res. 40(2), 342–354 (1992)
Desrochers, M., Lenstra, J.K., Savelsbergh, M.W., Soumis, F.: Vehicle routing with time windows: optimization and approximation. Veh. Routing: Methods Stud. 16, 65–84 (1988)
Feuerstein, E., Stougie, L.: On-line single-server dial-a-ride problems. Theoret. Comput. Sci. 268(1), 91–105 (2001)
Golden, B.L., Levy, L., Vohra, R.: The orienteering problem. Naval Res. Logist. 34(3), 307–318 (1987)
Irani, S., Lu, X., Regan, A.: On-line algorithms for the dynamic traveling repair problem. J. Sched. 7(3), 243–258 (2004)
Jaillet, P., Lu, X.: Online traveling salesman problems with rejection options. Networks 64(2), 84–95 (2014)
Karuno, Y., Nagamochi, H.: 2-approximation algorithms for the multi-vehicle scheduling problem on a path with release and handling times. Discrete Appl. Math. 129(2), 433–447 (2003)
Krumke, S.O., Megow, N., Vredeveld, T.: How to whack moles. In: Solis-Oba, R., Jansen, K. (eds.) WAOA 2003. LNCS, vol. 2909, pp. 192–205. Springer, Heidelberg (2004). doi:10.1007/978-3-540-24592-6_15
Nagamochi, H., Ohnishi, T.: Approximating a vehicle scheduling problem with time windows and handling times. Theoret. Comput. Sci. 393(1), 133–146 (2008)
Savelsbergh, M.W.: Local search in routing problems with time windows. Ann. Oper. Res. 4(1), 285–305 (1985)
Tan, K.C., Lee, L.H., Zhu, Q., Ou, K.: Heuristic methods for vehicle routing problem with time windows. Artif. Intell. Eng. 15(3), 281–295 (2001)
Thangiah, S.R.: Vehicle Routing with Time Windows Using Genetic Algorithms. Citeseer (1993)
Tsitsiklis, J.N.: Special cases of traveling salesman and repairman problems with time windows. Networks 22(3), 263–282 (1992)
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Azar, Y., Vardi, A. (2017). Dynamic Traveling Repair Problem with an Arbitrary Time Window. In: Jansen, K., Mastrolilli, M. (eds) Approximation and Online Algorithms. WAOA 2016. Lecture Notes in Computer Science(), vol 10138. Springer, Cham. https://doi.org/10.1007/978-3-319-51741-4_2
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