Abstract
An orientation of an undirected graph G is a directed graph obtained by replacing each edge {u,v} of G by exactly one of the arcs (u,v) or (v,u). In the min-sum k-paths orientation problem, the input is an undirected graph G and ordered pairs (s i ,t i ), where i ∈ {1,2,…,k}. The goal is to find an orientation of G that minimizes the sum over every i ∈ {1,2,…,k} of the distance from s i to t i .
In the min-sum k edge-disjoint paths problem the input is the same, however the goal is to find for every i ∈ {1,2,…,k} a path between s i and t i so that these paths are edge-disjoint and the sum of their lengths is minimum. Note that, for every fixed k ≥ 2, the question of NP-hardness for the min-sum k-paths orientation problem and the min-sum k edge-disjoint paths problem have been open for more than two decades. We study the complexity of these problems when k = 2.
We exhibit a PTAS for the min-sum 2-paths orientation problem. A by-product of this PTAS is a reduction from the min-sum 2-paths orientation problem to the min-sum 2 edge-disjoint paths problem. The implications of this reduction are: (i) an NP-hardness proof for the min-sum 2-paths orientation problem yields an NP-hardness proof for the min-sum 2 edge-disjoint paths problem, and (ii) any approximation algorithm for the min-sum 2 edge-disjoint paths problem can be used to construct an approximation algorithm for the min-sum 2-paths orientation problem with the same approximation guarantee and only an additive polynomial increase in the running time.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Han, J., Jahanian, F.: Impact of path diversity on multi-homed and overlay networks. In: DSN 2004, p. 29. IEEE Computer Society (2004)
Hassin, R., Megiddo, N.: On orientations and shortest paths. Linear Algebra Appl. 114–115, 589–602 (1989)
Ito, T., Miyamoto, Y., Ono, H., Tamaki, H., Uehara, R.: Route-enabling graph orientation problems. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 403–412. Springer, Heidelberg (2009)
Kammer, F., Tholey, T.: The k-disjoint paths problem on chordal graphs. In: Paul, C., Habib, M. (eds.) WG 2009. LNCS, vol. 5911, pp. 190–201. Springer, Heidelberg (2010)
Kobayashi, Y., Sommer, C.: On shortest disjoint paths in planar graphs. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 293–302. Springer, Heidelberg (2009)
Li, C., McCormick, T.S., Simich-Levi, D.: The complexity of finding two disjoint paths with min-max objective function. Discrete Appl. Math. 26(1), 105–115 (1989)
Robertson, N., Seymour, P.D.: Graph minors. XIII. The disjoint paths problem. J. Comb. Theory Ser. B 63(1), 65–110 (1995)
Vasudevan, V., Andersen, D.G., Zhang, H.: Understanding the AS-level path disjointness provided by multi-homing. Technical Report CMU-CS-07-141. Carnegie Mellon University (2007)
Yang, B., Zheng, S.Q.: Finding min-sum disjoint shortest paths from a single source to all pairs of destinations. In: Cai, J.-Y., Cooper, S.B., Li, A. (eds.) TAMC 2006. LNCS, vol. 3959, pp. 206–216. Springer, Heidelberg (2006)
Zhang, P., Zhao, W.: On the complexity and approximation of the min-sum and min-max disjoint paths problems. In: Chen, B., Paterson, M., Zhang, G. (eds.) ESCAPE 2007. LNCS, vol. 4614, pp. 70–81. Springer, Heidelberg (2007)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Fenner, T., Lachish, O., Popa, A. (2014). Min-Sum 2-Paths Problems. In: Kaklamanis, C., Pruhs, K. (eds) Approximation and Online Algorithms. WAOA 2013. Lecture Notes in Computer Science, vol 8447. Springer, Cham. https://doi.org/10.1007/978-3-319-08001-7_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-08001-7_1
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-08000-0
Online ISBN: 978-3-319-08001-7
eBook Packages: Computer ScienceComputer Science (R0)