Abstract
Given a number α with 0 < α < 1, a network G = (V, E) and two nodes s and t in G, we consider the problem of finding two disjoint paths P 1 and P 2 from s to t such that length(P1) ≤ length(P 2) and length(P 1) + α·length(P 2) is minimized. The paths may be node-disjoint or edge-disjoint, and the network may be directed or undirected. This problem has applications in reliable communication. We prove an approximation ratio \({1+\alpha} \over {2\alpha}\) for all four versions of this problem, and also show that this ratio cannot be improved for the two directed versions unless P = NP. We also present Integer Linear Programming formulations for all four versions of this problem. For a special case of this problem, we give a polynomial-time algorithm for finding optimal solutions.
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References
Harary, F.: Graph Theory. Addison-Wesley, Reading (1972)
Even, S.: Graph Algorithms. Computer Science, Rockville (1979)
Ford, L.R., Fulkerson, D.R.: Flows in Networks, Princeton (1962)
Li, C.L., McCormick, S.T., Simchi-Levi, D.: The Complexity of Finding Two Disjoint Paths with Min-Max Objective Function. Discrete Appl. Math. 26(1), 105–115 (1990)
Li, C.L., McCormick, S.T., Simchi-Levi, D.: Finding Disjoint Paths with Different Path-Costs: Complexity and Algorithms. Networks 22, 653–667 (1992)
Fortune, S., Hopcroft, J., Wyllie, J.: The directed subgraph homeomorphism problem. Theoret. Comput. Sci. 10, 111–121 (1980)
Yang, B., Zheng, S.Q., Katukam, S.: Finding Two Disjoint Paths in a Network with Min-Min Objective Function. In: Proceedings of the 15th IASTED International Conference on Parallel and Distributed Computing and Systems, pp. 75–80 (2003)
Yang, B., Zheng, S.Q., Lu, E.: Finding Two Disjoint Paths in a Network with Normalized α −-Min-Sum Objective Function (manuscript)
Perl, Y., Shiloach, Y.: Finding Two Disjoint Paths between Two Pairs of Vertices in a Graph. J. ACM 25(1), 1–9 (1978)
Shiloach, Y.: A polynomial solution to the undirected two paths problem. J. ACM 27(3), 445–456 (1980)
Suurballe, J.W.: Disjoint Paths in a Network. Networks 4, 125–145 (1974)
Suurballe, J.W., Tarjan, R.E.: A Quick Method for Finding Shortest Pairs of Disjoint Paths. Networks 14, 325–336 (1984)
Kleinberg, J.M.: Single-source unsplittable flow. In: Proceedings of the 37th Annual Symposium on foundations of Computer Science, pp. 68–77 (1996)
Fallah, F., Liao, S., Devadas, S.: Solving Covering Problems Using LPR-Based Lower Bounds. IEEE Transactions on Very Large Scale Integration (VLSI) Systems 8(1), 9–17 (2000)
Fallah, F., Devadas, S., Keutzer, K.: Functional Vector Generation for HDL Models Using Linear Programming and Boolean Satisfiability. IEEE Transactions on computer Aided Design of Integrated Circuits and Systems 20(8), 994–1002 (2001)
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Yang, B., Zheng, S.Q., Lu, E. (2005). Finding Two Disjoint Paths in a Network with Normalized α + -MIN-SUM Objective Function. In: Deng, X., Du, DZ. (eds) Algorithms and Computation. ISAAC 2005. Lecture Notes in Computer Science, vol 3827. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11602613_95
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DOI: https://doi.org/10.1007/11602613_95
Publisher Name: Springer, Berlin, Heidelberg
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