Approximability of Robust Network Design: The Directed Case

We consider robust network design problems where an uncertain traffic vector belonging to a polytope has to be dynamically routed to minimize either the network congestion or some linear reservation cost. We focus on the variant in which the underlying graph is directed. We prove that an O(?k) = O(n)-approximation can be obtained by solving the problem under static routing, where k is the number of commodities and n is the number of nodes. This improves previous results of Hajiaghayi et al. [SODA\u272005] and matches the ?(n) lower bound of Ene et al. [STOC\u272016] and the ?(?k) lower bound of Azar et al. [STOC\u272003]. Finally, we introduce a slightly more general problem version where some flow restrictions can be added. We show that it cannot be approximated within a ratio of k^{c/(log log k)} (resp. n^{c/(log log n)}) for some constant c. Making use of a weaker complexity assumption, we prove that there is no approximation within a factor of 2^{log^{1- ?} k} (resp. 2^{log^{1- ?} n}) for any ? > 0

Robust Two-Stage Packing into Designated and Multipurpose Bins

International audienceMultitype bin packing is a natural extension of the classical bin packing with applications to shipping using climate-controlled containers and plain dry containers. In transportation and other logistics applications there may be significant uncertainty with respect to the exact quantities of different variants of products (or item types) that may need to be shipped at the time when the containers and packaging are procured. In the current paper we model the problem as a robust two-stage two-item type bin packing problem. In the first stage bins of different types are acquired (e.g., reefer containers and dry containers). In the second stage the items are packed into bins. The bins that are secured in the first phase must allow for all of the items to be packed in the "worst-case" demand scenario. We first develop an algorithm for the robust two-stage two-item type bin packing problem with general item-number uncertainty sets and certain box uncertainty sets for item sizes (or equivalently two item sizes). We then consider the special case of identical (or unit) item sizes. In this special case we develop closed-form solutions for the optimal solution. Our closed-form solution reveals that it is optimal to use a number of multipurpose bins that is linear in the number of items. This is in contrast with solutions of the online and offline deterministic version of our problem that use at most one multipurpose bin. Finally, we consider computational methods that are efficient in practice for a generalization with unit item sizes but with an arbitrary number of item and bin types and arbitrary compatibility structures

On the approximability of robust network design

International audienceGiven the dynamic nature of traffic, we investigate the variant of robust network design where we have to determine the capacity to reserve on each link so that each demand vector belonging to a polyhedral set can be routed. The objective is either to minimize congestion or a linear cost. Routing is assumed to be fractional and dynamic (i.e., dependent on the current traffic vector). We first prove that the robust network design problem with minimum congestion cannot be approximated within any constant factor. Then, using the ETH conjecture, we get a Ω(log⁡n/log⁡log⁡n) lower bound for the approximability of this problem. This implies that the well-known O(log⁡n) approximation ratio established by Räcke in 2008 is tight. Using Lagrange relaxation, we obtain a new proof of the O(log⁡n) approximation. An important consequence of the Lagrange-based reduction and our inapproximability results is that the robust network design problem with linear reservation cost cannot be approximated within any constant ratio. This answers a long-standing open question of Chekuri (2007). We also give another proof of the result of Goyal et al. (2009) stating that the optimal linear cost under static routing can be Ω(log⁡n) more expensive than the cost obtained under dynamic routing. Finally, we show that even if only two given paths are allowed for each commodity, the robust network design problem with minimum congestion or linear cost is hard to approximate within some constant

Affine routing for robust network design

International audienceTaking into account the dynamic nature of traffic in telecommunication networks, the robust network design problem is to fix the edge capacities so that all demand vectors belonging to a polytope can be routed. While a common heuristic for this co-NP-hard problem is to compute, in polynomial time, an optimal static routing, affine routing can be used to obtain better solutions. It consists in restricting the routing to affinely depend on the demands. We show that a node-arc formulation is less conservative than an arc-path formulation. We also provide a cycle-based formulation that is equivalent to the node-arc formulation. To further reduce the solution's cost, several new formulations are obtained by relaxing flow conservation constraints and aggregating demands. As might be expected, aggregation allows us to reduce the size of formulations. A more striking result is that aggregation reduces the solution's cost

Programmation dynamique pour l'optimisation du clustering des configurations réseau

International audienceLes réseaux SDN (software-defined networking) permettent d'optimiser globalement la configuration des réseaux en fonction du trafic. Idéalement, le réseau devrait être reconfiguré de manière dynamique afin de permettre une meilleure utilisation des ressources. Néanmoins, en pratique, les changements de routes ne peuvent pas être trop fréquents à cause de la lente mise à jour des équipements et de la dynamicité individuelle des flux. Pour trouver un bon compromis entre fréquence de reconfiguration et l'efficacité de l'utilisation des ressources, une nouvelle approche d'ingénierie de trafic basée sur le clustering robuste a été proposée dans [SFC + 18]. L'idée principale est de regrouper des scénarios de trafic futurs présentants des similitudes de routage et temporel. Pour résoudre le problème de clustering robuste d'une manière optimale et rapide, nous proposons un nouvel algorithme basé sur la programmation dynamique. Nous comparons l'algorithme avec l'approche heuristique de [SFC + 18] et la résolution de l'ILP sur des instances réelles. Les résultats montrent que notre approche est efficace pour trouver la solution optimale et améliore les performances du routage par rapport à l'heuristique. Mots-clefs : Réseaux logiciels, routage robuste, calcul de chemins, programmation dynamiqu

Programmation dynamique pour l'optimisation du clustering des configurations réseau

International audienceLes réseaux SDN (software-defined networking) permettent d'optimiser globalement la configuration des réseaux en fonction du trafic. Idéalement, le réseau devrait être reconfiguré de manière dynamique afin de permettre une meilleure utilisation des ressources. Néanmoins, en pratique, les changements de routes ne peuvent pas être trop fréquents à cause de la lente mise à jour des équipements et de la dynamicité individuelle des flux. Pour trouver un bon compromis entre fréquence de reconfiguration et l'efficacité de l'utilisation des ressources, une nouvelle approche d'ingénierie de trafic basée sur le clustering robuste a été proposée dans [SFC + 18]. L'idée principale est de regrouper des scénarios de trafic futurs présentants des similitudes de routage et temporel. Pour résoudre le problème de clustering robuste d'une manière optimale et rapide, nous proposons un nouvel algorithme basé sur la programmation dynamique. Nous comparons l'algorithme avec l'approche heuristique de [SFC + 18] et la résolution de l'ILP sur des instances réelles. Les résultats montrent que notre approche est efficace pour trouver la solution optimale et améliore les performances du routage par rapport à l'heuristique. Mots-clefs : Réseaux logiciels, routage robuste, calcul de chemins, programmation dynamiqu