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22 April K. Andreas and J. Cole Smithsingle-path problem. Our algorithms for these two-path problems are based on relaxing acomplicating nonconvex constraint and providing a (hopefully tight) series of disjunctiveconvex approximations to the problem. For the multiple path problem, we instead turn toa path-based formulation instead of the arc-based formulation for the two-path problem, andreplace the reliability constraint with a set of linear cover constraints. We propose the useof a branch-and-price-and-cut framework to solve these problems. Our future research willinvolve providing the details of the branch-and-cut-and-price algorithm, and stating computationalresults comparing the efficacy of using our algorithm on model formulations.AcknowledgmentsThe authors gratefully acknowledge the support of the Office of Naval Research under GrantNumber N00014-03-1-0510 and the Air Force Office of Scientific Research under Grant NumberF49620-03-1-0377.References[1] A.K. Andreas and J.C. Smith. Mathematical Programming Algorithms for Two-Path Routing Problems withReliability Considerations. Working Paper, Tucson, AZ, 2005.[2] J.F. Bard and J.L. Miller. Probabilistic Shortest Path Problems with Budgetary Constraints. Computers and OperationsResearch, 16(2):145–159, 1989.[3] C. Barnhart, C.A. Hane, and P.H. Vance. Using Branch-and-Price-and-Cut to Solve Origin-Destination IntegerMulticommodity Flow Problems. Operations Research, 48(2):318–326, 2000.[4] A.A. Elimam and D. Kohler. Case Study: Two Engineering Applications of a Constrained Shortest-Path Model.European Journal of Operational Research, 103:426–438, 1997.[5] S. Fortune, J. Hopcroft, and J. Wyllie. The Directed Subgraph Homeomorphism Problem. Theoretical ComputerScience, 10:111–121, 1980.[6] G. Li, D. Wang, C. Kalmanek, and R. Doverspike. Efficient Distributed Path Selection for Shared RestorationConnections. IEEE INFOCOM 2002, 1:140–149, 2002.[7] Y. Liu and D. Tipper. Successive Survivable Routing for Node Failures. GLOBECOM 2001 - IEEE GlobalTelecommunications Conference 1:2093–2097, 2001.[8] H.D. Sherali and C.H. Tuncbilek. A Global Optimization Algorithm for Polynomial Programming ProblemsUsing a Reformulation-Linearization Technique. Journal of Global Optimization, 2:101–112, 1992.[9] J.W. Suurballe. Disjoint Paths in a Network. Networks, 4:125–145, 1974.[10] D. Xu, C. Qiao, and Y. Xiong. An Ultra-fast Shared Path Protection Scheme-Distributed Partial InformationManagement, Part II. In Proceedings of the 10th IEEE International Conference on Network Protocols, pages 344–353, 2002.[11] J.R. Yee and F.Y.S. Lin. A Routing Algorithm for Virtual Circuit Data Networks with Multiple Sessions PerO-D Pair. Networks, 22:185–208, 1992.[12] M. Zabarankin, S. Uryasev, and P. Pardalos. Optimal Risk Path Algorithms. In Cooperative Control and Optimization,pages 273–298, Dordrecht; Boston: Kluwer Academic Publishers, 2001.