Abstract
A weakness of next-hop routing is that following a link or router failure there may be no routes between some source-destination pairs, or packets may get stuck in a routing loop as the protocol operates to establish new routes. In this article, we address these weaknesses by describing mechanisms to choose alternate next hops. Our first contribution is to model the scenario as the following tree augmentation problem. Consider a mixed graph where some edges are directed and some undirected. The directed edges form a spanning tree pointing towards the common destination node. Each directed edge represents the unique next hop in the routing protocol. Our goal is to direct the undirected edges so that the resulting graph remains acyclic and the number of nodes with outdegree two or more is maximized. These nodes represent those with alternative next hops in their routing paths. We show that tree augmentation is NP-hard in general and present a simple \(\frac{1}{2}\)-approximation algorithm. We also study 3 special cases. We give exact polynomial-time algorithms for when the input spanning tree consists of exactly 2 directed paths or when the input graph has bounded treewidth. For planar graphs, we present a polynomial-time approximation scheme when the input tree is a breadth-first search tree. To the best of our knowledge, tree augmentation has not been previously studied.
Similar content being viewed by others
References
Arnborg S, Corneil D, Proskurowski A (1987) Complexity of finding embeddings in a \(k\)-tree. SIAM J Algebraic Discrete Methods 8:277–284
Baker B (1994) Approximation algorithms for NP-complete problems on planar graphs. J ACM 41:153–180
Bodlaender H (1993) A tourist guide through treewidth. Acta Cybern 11:1–23
Bodlaender H (1996) A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J Comput 25(6):1305–1317
Courcelle B, Mosbah M (1992) Monadic second-order evaluations on tree-decomposable graphs. In: Schmidt Gunther, Berghammer Rudolf (eds) Graph-theoretic concepts in computer science, vol 570., Lecture Notes in Computer ScienceSpringer, Berlin/Heidelberg, pp 13–24
Garey M, Johnson D (1979) Computers and intractability: a guide to the theory of NP-completeness. W. H. Freeman and Co., San Francisco, CA
Goyal M, Soperi M, Baccelli E, Choudhury G, Shaikh A, Hosseini H, Trivedi K (2012) Improving convergence speed and scalability in OSPF: a survey. IEEE Commun Surv Tutor 14(2):443–463
Hedrick C (1988) Routing information protocol. RFC 1058
Moy J (1998) OSPF Version 2. RFC 2178
Oran D (1990) OSI IS-IS intra-domain routing protocol. RFC 1142
Raj A, Ibe O (2007) A survey of IP and multiprotocol label switching fast reroute schemes. Comput Netw 51(8):1882–1907
Vo HQ, Lysne O, Kvalbein A (2012) Permutation routing for increased robustness in IP networks. In: Proceedings of the 11th international IFIP TC 6 conference on networking, vol I, pp 217–231
Acknowledgments
This material is based upon work supported by the National Science Foundation, under Grant No. CCF-0964037. Sean Kennedy is partially supported by a postdoctoral fellowship from the Natrual Sciences and Engineering Research Council of Canada.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Borradaile, G., Kennedy, W.S., Wilfong, G. et al. Improving robustness of next-hop routing. J Comb Optim 31, 1206–1220 (2016). https://doi.org/10.1007/s10878-014-9818-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-014-9818-x