Abstract
We give a randomized algorithm for maximum edge-disjoint paths problem (MEDP) and the minimal total length of MEDP, if the graphs are planar and all terminals lie on the outer face in the order s 1, s 2, ...s k , t k , t k − 1, ...t 1. Moreover, if the degree of the graph is bounded by 3, the algorithm becomes deterministic and can also output the number of optimal solutions. On the other hand, we prove that the counting version of these problems are #P-hard even if restricted to planar graphs with maximum degree 3.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Brandes, U., Neyer, G., Wagner, D.: Edge-disjoint paths in planar graphs with short total length. Technical Reports (1996)
Fishier, M.E.: Statistical mechanics of dimers on a plane lattice. Phys. Rev. 124, 1664–1672 (1961)
Kasteleyn, P.W.: The statistics of dimers on a lattice. Physica 27, 1209–1225 (1961)
Kasteleyn, P.W.: Graph theory and crystal physics. In: Harary, F. (ed.) Graph Theory and Theoretical Physics, pp. 43–110. Academic Press, London (1967)
Middendorf, M., Pfeiffer, F.: On the complexity of the disjoint paths problem. Combinatorica 13, 97–107 (1993)
Papadimitriou, C.: Computational Complexity. Addison-Wesley, Reading (1994)
Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficieny. Springer, Heidelberg (2003)
Sebő, A.: Integer plane multiflows with a fixed number of demands. J. Comb. Theory Ser. B 59, 163–171 (1993)
Temperley, H.N.V., Fishier, M.E.: Dimer problems in statistical mechanics - An exact result. Philosophical Magazine 6, 1061–1063 (1961)
Wagner, D., Weihe, K.: A linear-time algorithm for edge-disjoint paths in planar graphs. Combinatorica 15, 135–150 (1995)
Vadhan, S.P.: The complexity of counting in sparse, regular, and planar graphs. SIAM Journal on Computing 31, 398–427 (2001)
Valiant, L.G.: The complexity of enumeration and reliability problems. SIAM Journal on Computing 8, 410–421 (1979)
Valiant, L.G.: Quantum circuits that can be simulated classically in polynomial time. SIAM Journal on Computing 31, 1229–1254 (2002)
Valiant, L.G.: Holographic algorithms (extended abstract). In: FOCS, 2004, pp. 306–315 (2004)
Valiant, L.G.: Accidental Algorithms. In: FOCS, 2006, pp. 509–517 (2006)
Zhao, W.-B., Xia, M.: #3-Regular Bipartite Planar Vertex Cover is #P-Complete. In: Cai, J.-Y., Cooper, S.B., Li, A. (eds.) TAMC 2006. LNCS, vol. 3959, pp. 356–364. Springer, Heidelberg (2006)
Zhang, P., Zhao, W.: On the complexity and approximation fo the min-sum and min-max disjoint paths problems. Manuscripts.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Xia, M. (2007). Maximum Edge-Disjoint Paths Problem in Planar Graphs. In: Cai, JY., Cooper, S.B., Zhu, H. (eds) Theory and Applications of Models of Computation. TAMC 2007. Lecture Notes in Computer Science, vol 4484. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72504-6_51
Download citation
DOI: https://doi.org/10.1007/978-3-540-72504-6_51
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-72503-9
Online ISBN: 978-3-540-72504-6
eBook Packages: Computer ScienceComputer Science (R0)