Abstract
Given four distinct vertices s 1,s 2,t 1, and t 2 of a graph G, the 2-disjoint paths problem is to determine two disjoint paths, p 1 from s 1 to t 1 and p 2 from s 2 to t 2, if such paths exist. Disjoint can mean vertex- or edge-disjoint.
Both, the edge- and the vertex-disjoint version of the problem, are \(\mathcal{NP}\)-hard in the case of directed graphs. For undirected graphs, we show that the O(mn)-time algorithm of Shiloach can be modified so as to solve the 2-(vertex-)disjoint paths problem in O(n+mα(m,n)) time, where m is the number of edges in G, n is the number of vertices in G, and α denotes the inverse of the Ackermann function. Our result also improves the running time for the 2-edge-disjoint paths problem on undirected graphs as well as the running times for the decision versions of the 2-vertex- and the 2-edge-disjoint paths problem on dags.
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
Aggarwal, A., Kleinberg, J., Williamson, D.P.: Node-disjoint paths on the mesh and a new trade-off in VLSI layout. SIAM J. Comput. 29, 1321–1333 (2000)
Bang-Jensen, J., Gutin, G.: Digraphs: Theory, Algorithms and Applications. Springer, London (2001)
Dinitz, Y., Westbrook, J.: Maintaining the classes of 4-edge-connectivity in a graph on-line. Algorithmica 20, 242–276 (1998)
Even, S., Itai, A., Shamir, A.: On the complexity of timetable and multicommodity flow problems. SIAM J. Comput. 5, 691–703 (1976)
Fortune, S., Hopcroft, J., Wyllie, J.: The directed subgraph homeomorphism problem. Theoret. Comput. Sci. 10, 111–121 (1980)
Gopalakrishnan, C.P., Pandu Rangan, C.: Edge-disjoint paths in permutation graphs. Discuss. Math. Graph Theory 15, 59–72 (1995)
Gustedt, J.: The general two-path problem in time O(m log n) (extended abstract), Report No. 394/1994, TU Berlin, FB Mathematik (1994)
Jungnickel, D.: Graphen, Netzwerke und Algorithmen, B. I. Wissenschaftsverlag, Mannheim (1994)
Kanevsky, A., Tamassia, R., Di Battista, G., Chen, J.: On-line maintenance of the four-connected components of a graph. In: Proc. 32nd Annual IEEE Symposium on Foundations of Computer Science (FOCS 1991), pp. 793–801 (1991)
Karp, R.M.: On the computational complexity of combinatorial problems. Networks 5, 45–68 (1975)
Khuller, S., Mitchell, S.G., Vazirani, V.V.: Processor efficient parallel algorithms for the two disjoint paths problem and for finding a Kuratowski homeomorph. SIAM J. Comput. 21, 486–506 (1992)
Korach, E., Tal, A.: General vertex disjoint paths in series-parallel graphs. Discrete Appl. Math. 41, 147–164 (1993)
Lucchesi, C.L., Giglio, M.C.M.T.: On the irrelevance of edge orientations on the acyclic directed two disjoint paths problem, IC Technical Report DCC-92-03, Universidade Estadual de Campinas, Instituto de Computação (1992)
Lynch, J.F.: The equivalence of theorem proving and the interconnection problem. (ACM) SIGDA Newsletter 5, 31–36 (1975)
Ohtsuki, T.: The two disjoint path problem and wire routing design. In: Saito, N., Nishizeki, T. (eds.) Graph Theory and Algorithms. LNCS, vol. 108, pp. 207–216. Springer, Heidelberg (1981)
Perković, L., Reed, B.: An improved algorithm for finding tree decompositions of small width. International Journal of Foundations of Computer Science (IJFCS) 11, 365–372 (2000)
Perl, Y., Shiloach, Y.: Finding two disjoint paths between two pairs of vertices in a graph. J. ACM 25, 1–9 (1978)
Robertson, N., Seymour, P.D.: Graph minors. XIII. The disjoint paths problem. J. Comb. Theory, Ser. B 63, 65–110 (1995)
Scheffler, P.: A practical linear time algorithm for disjoint paths in graphs with bounded tree-width, Report No. 396/1994, TU Berlin, FB Mathematik (1994)
Schrijver, A.: A group-theoretical approach to disjoint paths in directed graphs. CWI Quarterly 6, 257–266 (1993)
Schrijver, A.: Combinatorial optimization – Polyhedra and Efficiency, vol. C. Springer, Berlin (2002)
Schwill, A.: Nonblocking graphs: greedy algorithms to compute disjoint paths. In: Choffrut, C., Lengauer, T. (eds.) STACS 1990. LNCS, vol. 415, pp. 250–262. Springer, Heidelberg (1990)
Seymour, P.D.: Disjoint paths in graphs. Discrete Math. 29, 293–309 (1980)
Shiloach, Y.: A polynomial solution to the undirected two paths problem. J. ACM 27, 445–456 (1980)
Suurballe, J.W., Tarjan, R.E.: A quick method for finding shortest pairs of disjoint paths. Networks 14, 325–336 (1984)
Thomassen, C.: 2-linked graphs. Europ. J. Combinatorics 1, 371–378 (1980)
Watkins, M.E.: On the existence of certain disjoint arcs in graphs. Duke Math. J. 35, 231–246 (1968)
Williamson, S.G.: Depth-first search and Kuratowski subgraphs. J. ACM 31, 681–693 (1984)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Tholey, T. (2004). Solving the 2-Disjoint Paths Problem in Nearly Linear Time. In: Diekert, V., Habib, M. (eds) STACS 2004. STACS 2004. Lecture Notes in Computer Science, vol 2996. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24749-4_31
Download citation
DOI: https://doi.org/10.1007/978-3-540-24749-4_31
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-21236-2
Online ISBN: 978-3-540-24749-4
eBook Packages: Springer Book Archive