Network virtualization promises a high flexibility by decoupling services from the underlying substrate network and allowing the virtual network to adapt to the needs of the service, e.g., by migrating servers or/and parts of the network. We study a system (e.g., a gaming application) where network virtualization is used to support thin client applications for mobile devices to improve their QoS. To deal with the dynamics of both the mobile clients as well as the ability to migrate services closer to the client location we advocate, in this paper, the use of competitive analysis. After identifying the parameters that characterize the cost-benefit tradeoff for this kind of application we propose an online migration strategy. The strength of the strategy is that it is robust with regards to any arbitrary request access pattern. In particular, it is close to the optimal offline algorithm that knows the access pattern in advance.In this paper we present both an optimal offline algorithm based on dynamic programming techniques to find the best migration paths for a given request sequence, and a O(µ log n)-competitive migration strategy MIG where µ is the ratio between maximal and minimal link capacity in the substrate network for a simplified model. This is almost optimal for small µ, as we also show that there are networks where no online algorithm can achieve a ratio * Part of this work was performed within the 4WARD project, which is funded by the European Union in the 7th Framework Programme (FP7), the Virtu project, funded by NTT DOCOMO Euro-Labs, and the Collaborative Networking project, funded by Deutsche Telekom AG. We would like to thank our colleagues in these projects for many fruitful discussions. M. Bienkowski is supported by MNiSW grant number N N206 1723 33, 2007-2010. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. below Ω(log n/ log log n). In contrast, the optimal solution without migration can only achieve a competitive ratio that is linear in the network diameter. Our simulations indicate that the competitive ratio of MIG is robust to the network size, and that the ratio is small if the request dynamics are limited and the requests are correlated.
The Joint Replenishment Problem (JRP) deals with optimizing shipments of goods from a supplier to retailers through a shared warehouse. Each shipment involves transporting goods from the supplier to the warehouse, at a fixed cost C, followed by a redistribution of these goods from the warehouse to the retailers that ordered them, where transporting goods to a retailer ρ has a fixed cost c ρ . In addition, we incur waiting costs for each order, possibly an arbitrary non-decreasing function of time, different for each order. The objective is to minimize the overall cost of satisfying all orders, namely the sum of all shipping and waiting costs.JRP has been well studied in Operations Research and, more recently, in the area of approximation algorithms. For arbitrary waiting cost functions, the best known approximation ratio is 1.8. This ratio can be reduced to ≈ 1.574 for the JRP-D model, where there is no cost for waiting but orders have deadlines. As for hardness results, it is known that the problem is APXhard and that the natural linear program for JRP has integrality gap at least 1.245. Both results hold even for JRP-D. In the online scenario, the best lower and upper bounds on the competitive ratio are 2.64 and 3, respectively. The lower bound of 2.64 applies even to the restricted version of JRP, denoted JRP-L, where the waiting cost function is linear.We provide several new approximation results for JRP. In the offline case, we give an algorithm with ratio ≈ 1.791, breaking the barrier of 1.8. We also show that the integrality gap of the linear program for JRP-L is at * Research partially supported by NSF grants CCF-1217314 and OISE-1157129, MNiSW grant no. N N206 368839, 2010-2013 least 12/11 ≈ 1.09. In the online case, we show a lower bound of ≈ 2.754 on the competitive ratio for JRP-L (and thus JRP as well), improving the previous bound of 2.64. We also study the online version of JRP-D, for which we prove that the optimal competitive ratio is 2. IntroductionThe Joint Replenishment Problem (JRP) deals with optimizing shipments of goods from a supplier to a set R of retailers through a shared warehouse. Over time, retailers issue orders for items. All ordered items must be subsequently shipped to the retailers, although some shipments can be delayed, in order to aggregate orders into fewer shipments to reduce cost.Specifically, for each ρ ∈ R we are given the cost c ρ of transporting goods from the warehouse to ρ. We are also given the cost C of transporting goods from the supplier to the warehouse. A shipment of goods from the supplier to a subset S ⊆ R of retailers involves first shipping them to the warehouse and then redistributing them to all retailers in S, at cost equal C + ρ∈S c ρ . Note that this cost is independent of the set of items shipped. The waiting cost of an item π ordered at time a and delivered at time t ≥ a is given by a function h(t), possibly dependent on π, where we assume that the values of h(t) are non-decreasing with t. The objective is to minimize the overall cost of s...
In a (randomized) oblivious routing scheme the path chosen for a request between a source s and a target t is independent from the current traffic in the network. Hence, such a scheme consists of probability distributions over s − t paths for every source-target pair s, t in the network.In a recent result [11] it was shown that for any undirected network there is an oblivious routing scheme that achieves a polylogarithmic competitive ratio with respect to congestion. Subsequently, Azar et al.[4] gave a polynomial time algorithm that for a given network constructs the best oblivious routing scheme, i.e. the scheme that guarantees the best possible competitive ratio. Unfortunately, the latter result is based on the Ellipsoid algorithm; hence it is unpractical for large networks.In this paper we present a combinatorial algorithm for constructing an oblivious routing scheme that guarantees a competitive ratio of O(log 4 n) for undirected networks. Furthermore, our approach yields a proof for the existence of an oblivious routing scheme with competitive ratio O(log 3 n), which is much simpler than the original proof from [11].
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