Logarithmic inapproximability results for the minimum shortest path routing conflict problem

Nowadays most data networks use shortest path protocols such as OSPF or IS-IS to route traffic. Given administrative routing lengths for the links of a network, all data packets are sent along shortest paths with respect to these lengths from their source to their destination. One of the most fundamental problems in planning shortest path networks is to decide whether a given set S of routing paths forms a valid routing and, if this is not the case, to find a small subset R of paths that cannot occur together in any valid routing. In this paper we show that it is NP-hard to approximate the minimal size or the minimal weight of a shortest path conflict $R\subseteq S$ by a factor less than $c\log |S|$ for some $c>0$

Algorithms for Inverse Optimization Problems

We study inverse optimization problems, wherein the goal is to map given solutions to an underlying optimization problem to a cost vector for which the given solutions are the (unique) optimal solutions. Inverse optimization problems find diverse applications and have been widely studied. A prominent problem in this field is the inverse shortest path (ISP) problem [D. Burton and Ph.L. Toint, 1992; W. Ben-Ameur and E. Gourdin, 2004; A. Bley, 2007], which finds applications in shortest-path routing protocols used in telecommunications. Here we seek a cost vector that is positive, integral, induces a set of given paths as the unique shortest paths, and has minimum l_infty norm. Despite being extensively studied, very few algorithmic results are known for inverse optimization problems involving integrality constraints on the desired cost vector whose norm has to be minimized. Motivated by ISP, we initiate a systematic study of such integral inverse optimization problems from the perspective of designing polynomial time approximation algorithms. For ISP, our main result is an additive 1-approximation algorithm for multicommodity ISP with node-disjoint commodities, which we show is tight assuming P!=NP. We then consider the integral-cost inverse versions of various other fundamental combinatorial optimization problems, including min-cost flow, max/min-cost bipartite matching, and max/min-cost basis in a matroid, and obtain tight or nearly-tight approximation guarantees for these. Our guarantees for the first two problems are based on results for a broad generalization, namely integral inverse polyhedral optimization, for which we also give approximation guarantees. Our techniques also give similar results for variants, including l_p-norm minimization of the integral cost vector, and distance-minimization from an initial cost vector

A propos de la difficulté du routage égal par plus courts chemins

Les réseaux de communications sont devenus très complexes de nos jours. Cependant, depuis leur tout début avec ARPANET, les chercheurs et les ingénieurs ont conçu plusieurs protocoles aux divers équipements de ces réseaux afin de réduire cette complexité et d'atteindre la meilleure performance possible du réseau. OSPF ("Open Shortest Path First") est l'un des plus anciens protocoles qui ont tenu bon avec cette évolution. Il appartient à la sous-famille des protocoles de routage. Le but d'un protocol de routage est de rendre chaque routeur capable de décider, à chaque paquet recu, le prochain routeur voisin à qui ce paquet doit etre transmis. Quand le protocole de routage activé sur le réseau est OSPF, tout les paquets suivent des plus courts chemins pondérés, ou les poids (à voir comme des longueurs) sont fixés par l'administrateur réseau. Quand ces poids sont définis de manière à avoir plusieurs plus courts chemins pour une paire de noeuds, le routage dépendra de la règle qui est implémentée avec OSPF. Il y a plusieurs règles permettant de balancer le trafic entre les différents plus courts chemins, et l'une des plus connues est ECMP ("Equal Cost Multiple Path"): un noeud qui a plusieurs liens sortant sur le plus court chemin envers une destination d divisera le trafic qui lui arrive et qui est destiné à d de manière égale entre tous les chemins. Si le trafic nécessite de ne pas être partagé entre différents chemins, les poids doivent etre affectés de manière à ce que pour chaque paire de noeuds il y ait un unique plus court chemin. Afin de comprendre la difficulté du routage OSPF, nous avons regardé un problème assez simple, mais qui reste fondamental, où le but est de maximiser le débit quand une seule et unique paire de noeuds com- munique des données sur le réseau. A notre meilleure connaissance, ce problème a été prouvé NP-complet, utilisant une réduction au problème de la couverture d'un ensemble, et aucune approximation, avec garantie non trivial, n'a été proposée jusqu'à présent. Nous montrons, en utilisant une réduction différente que ce problème ne peut être approximée à un facteur constant et nous donnons une approximation qui dépend de la plus longue distance du graphe

On the Hardness of Equal Shortest Path Routing

International audienceIn telecommunication networks packets are carried from a source s to a destination t on a path that is determined by the underlying routing protocol. Most routing protocols belong to the class of shortest path routing protocols. In such protocols, the network operator assigns a length to each link. A packet going from s to t follows a shortest path according to these lengths. For better protection and efficiency, one wishes to use multiple (shortest) paths between two nodes. Therefore the routing protocol must determine how the traffic from s to t is distributed among the shortest paths. In the protocol called ospf-ecmp (for Open Shortest Path First -Equal Cost Multiple Path) the traffic incoming at every node is uniformly balanced on all outgoing links that are on shortest paths. In that context, the operator task is to determine the "best" link lengths, toward a goal such as maximizing the network throughput for given link capacities. In this work, we show that the problem of maximizing even a single commodity flow for the ospf- ecmp protocol cannot be approximated within any constant factor ratio. Besides this main theorem, we derive some positive results which include polynomial-time approximations and an exponential-time exact algorithm. We also prove that despite their weakness, our approximation and exact algorithms are, in a sense, the best possible

On the complexity of equal shortest path routing

International audienceIn telecommunication networks packets are carried from a source s to a destination t on a path determined by the underlying routing protocol. Most routing protocols belong to the class of shortest path routing protocols. In such protocols, the network operator assigns a length to each link. A packet going from s to t follows a shortest path according to these lengths. For better protection and efficiency, one wishes to use multiple (shortest) paths between two nodes. Therefore the routing protocol must determine how the traffic from s to t is distributed among the shortest paths. In the protocol called OSPF-ECMP (for Open Shortest Path First-Equal Cost Multiple Path) the traffic incoming at every node is uniformly balanced on all outgoing links that are on shortest paths. In that context, the operator task is to determine the " best " link lengths, toward a goal such as maximizing the network throughput for given link capacities. In this work, we show that the problem of maximizing even a single commodity flow for the OSPF-ECMP protocol cannot be approximated within any constant factor ratio. Besides this main theorem, we derive some positive results which include polynomial-time approximations and an exponential-time exact algorithm

Open Shortest Path First Routing Under Random Early Detection

This is the peer reviewed version of the following article: Liu, J. and Dimitrov, S. (2018), Open shortest path first routing under random early detection. NETWORKS, 71: 120-135., which has been published in final form at https://doi.org/10.1002/net.21792. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions.In this article, we consider a variant of Open Shortest Path First (OSPF) routing that accounts for Random Early Detection (RED), an Active Queue Management method for backbone networks. In the version of OSPF we consider in this article we only require a single network path be available between each origin and destination, a simplification of the OSPF protocol. We formulate a mixed integer non‐linear program to determine the data paths, referred to as a routing policy. We prove that determining an optimal OSPF routing policy that accounts for RED is NP‐Hard. Furthermore, in order for the generated routing policies to be real‐world implementable, referred to as realizable, we must determine weights for all arcs in the network such that solving the all‐pairs shortest path problem using these weights reproduces the routing policies. We show that determining if a set of all‐pairs routes is realizable is also NP‐Hard. Fortunately, using traffic data from three real‐world backbone networks, we are able to find realizable routing policies for these networks that account for RED, using an off‐the‐shelf solver, and policies found perform better than those used in each network at the time the data was collected.This work was partially funded by the Natural Sciences and Engineering Research Council of Canada (NSERC)

An origin-based model for unique shortest path routing

Link weights are the main parameters of shortest path routing protocols, the most commonly used protocols for IP networks. The problem of optimally setting link weights for unique shortest path routing is addressed. Due to the complexity of the constraints involved, there exist challenges to formulate the problem in such a way based on which a more efficient solution algorithm than the existing ones may be developed. In this paper, an exact formulation is first introduced and then mathematically proved correct. It is further illustrated that the formulation has advantages over a prior one in terms of both constraint structure and model size for a proposed decomposition method to solve the problem

Sur la complexité du routage OSPF

International audienceCe travail montre que dans un réseau (général) où le protocole de routage est OSPF avec la stratégie d'équilibrage de charge ECMP, le problème qui consiste à maximiser un flot simple d'une source vers un puits ne peut être approché à une constante près