Online k-Server Routing Problems

Bonifaci, Vincenzo; Stougie, Leen

doi:10.1007/s00224-008-9103-4

Cited by 16 publications

(9 citation statements)

References 26 publications

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“…For multiple-server problems, we consider speed and server augmentation (without precedence and capacity constraints); we show that our algorithm is 1 + 1 − m − 1 / + − 1 / -competitive, where m is the number of online servers (the offline has a single server) and is a measure of the problem data. The reader may constrast our result with the vehicle augmentation result of Bonifaci and Stougie (2007). When the online algorithm has m servers and the offline algorithm has m * m servers, the authors give an online algorithm that is 1 + 1 + 1/2 m/m * −1 -competitive.…”

Section: Our Contributionsmentioning

confidence: 99%

“…As mentioned previously, Ascheuer et al (2000) give a 2-competitive online algorithm for the online dial-a-ride problem with multiple servers and capacity constraints. Bonifaci and Stougie (2007) study the online TSP with m salesmen. For the case where all cities are on the real line, these authors give an asymptotically (as m → ) optimal online algorithm.…”

Section: Literature Reviewmentioning

confidence: 99%

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Generalized Online Routing: New Competitive Ratios, Resource Augmentation, and Asymptotic Analyses

Jaillet

Wagner

2008

Operations Research

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We consider online routing optimization problems where the objective is to minimize the time needed to visit a set of locations under various constraints; the problems are online because the set of locations are revealed incrementally over time. We consider two main problems: (1) the online traveling salesman problem (TSP) with precedence and capacity constraints, and (2) the online TSP with m salesmen. For both problems we propose online algorithms, each with a competitive ratio of 2; for the m-salesmen problem, we show that our result is best-possible. We also consider polynomialtime online algorithms. We then consider resource augmentation, where we give the online servers additional resources to offset the powerful offline adversary advantage. In this way, we address a main criticism of competitive analysis. We consider the cases where the online algorithm has access to faster servers, servers with larger capacities, additional servers, and/or advanced information. We derive improved competitive ratios. We also give lower bounds on the competitive ratios under resource augmentation, which in many cases are tight and lead to best-possible results. Finally, we study online algorithms from an asymptotic point of view. We show that, under general stochastic structures for the problem data, unknown and unused by the online player, the online algorithms are almost surely asymptotically optimal. Furthermore, we provide computational results that show that the convergence can be very fast.

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Section: Our Contributionsmentioning

confidence: 99%

Section: Literature Reviewmentioning

confidence: 99%

Generalized Online Routing: New Competitive Ratios, Resource Augmentation, and Asymptotic Analyses

Jaillet

Wagner

2008

Operations Research

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“…From the impossibility side, Feuerstein and Stougie [22] gave a lower bound for the TRP (that also holds already for a line) of 1 + √ 2 > 2.414 and the bound of 7/3 for randomized algorithms was presented by Krumke et al [33]. For the variant of the TRP with multiple servers, the deterministic lower bound is only 2 [14] (it holds for any number of servers). Clearly, all these lower bounds hold also for any variant of DARP.…”

Section: Previous Workmentioning

confidence: 99%

“…The phase-based algorithm Interval extends in a straightforward fashion to the DARP problem with an arbitrary assumption on the server capacity, both for the preemptive and non-preemptive variants: all the details of the solved problem are encapsulated in the computations of auxiliary schedules [33]. In the same manner, Interval can be enhanced to handle multiple servers [14]. Although this was not explicitly stated in [28], the algorithm Plan-And-Commit can be extended in the same way.…”

Section: Previous Workmentioning

confidence: 99%

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

Bienkowski,

Kraska,

Liu

2021

Preprint

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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.

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“…Jaillet and Wagner [24] have proposed a centralized deterministic algorithm with a competitive ratio of 2 (the best possible solution). Bonifaci and Stougie [11] consider a variant of this problem in which the cost of an algorithm is measured by the time when the last request is visited and the servers are not required to return to the depot. For this problem, when the online algorithm has k servers and the offline algorithm used to define the competitive ratio has only k ⋆ servers (k ⋆ ≤ k), they propose a centralized deterministic online algorithm with competitive ratio 1 + 1 + 1/2 ⌊k/k ⋆ ⌋−1 .…”

Section: Related Workmentioning

confidence: 99%

Distributed Multi-Depot Routing without Communications

Hwang¹,

Jaillet²,

Zhou³

2014

Preprint

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We consider and formulate a class of distributed multi-depot routing problems, where servers are to visit a set of requests, with the aim of minimizing the total distance travelled by all servers. These problems fall into two categories: distributed offline routing problems where all the requests that need to be visited are known from the start; distributed online routing problems where the requests come to be known incrementally. A critical and novel feature of our formulations is that communications are not allowed among the servers, hence posing an interesting and challenging question: what performance can be achieved in comparison to the best possible solution obtained from an omniscience planner with perfect communication capabilities? The worst-case (over all possible request-set instances) performance metrics are given by the approximation ratio (offline case) and the competitive ratio (online case).Our first result indicates that the online and offline problems are effectively equivalent: for the same request-set instance, the approximation ratio and the competitive ratio differ by at most an additive factor of 2, irrespective of the release dates in the online case. Therefore, we can restrict our attention to the offline problem. For the offline problem, we show that the approximation ratio given by the Voronoi partition is m (the number of servers). For two classes of depot configurations, when the depots form a line and when the ratios between the distances of pairs of depots are upper bounded by a sublinear function f (m) (i.e., f (m) = o(m)), we give partition schemes with sublinear approximation ratios O(log m) and Θ(f (m)) respectively. We also discuss several interesting open problems in our formulations: in particular, how our initial results (on the two deliberately chosen classes of depots) shape our conjecture on the open problems.

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Online k-Server Routing Problems

Cited by 16 publications

References 26 publications

Generalized Online Routing: New Competitive Ratios, Resource Augmentation, and Asymptotic Analyses

Generalized Online Routing: New Competitive Ratios, Resource Augmentation, and Asymptotic Analyses

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

Distributed Multi-Depot Routing without Communications

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