An approximation algorithm for the disjoint paths problem in even-degree planar graphs

In joint work with Eva Tardos in 1995, we asked whether it was possible to obtain a polynomial-time, polylogarithmic approximation algorithm for the disjoint paths problem in the class of all even-degree planar graphs. This paper answers the question in the affirmative, by providing such an algorithm. The algorithm builds on recent work of C. Chekuri et al. (2004, 2005), who considered muting problems in planar graphs where each edge can carry up to two paths.

[1]  William R. Pulleyblank,et al.  Two Steiner tree packing problems , 1995, STOC '95.

[2]  Yuval Rabani,et al.  Improved bounds for all optical routing , 1995, SODA '95.

[3]  Richard M. Karp,et al.  Global Wire Routing in Two-Dimensional Arrays (Extended Abstract) , 1983, FOCS.

[4]  Lisa Zhang,et al.  Logarithmic hardness of the undirected edge-disjoint paths problem , 2006, JACM.

[5]  Raymond E. Miller,et al.  Complexity of Computer Computations , 1972 .

[6]  Ronitt Rubinfeld,et al.  Short paths in expander graphs , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[7]  Paul D. Seymour,et al.  Graph minors. V. Excluding a planar graph , 1986, J. Comb. Theory B.

[8]  J. M. Sek.  On embedding trees into uniformly convex Banach spaces , 1999 .

[9]  András Frank.  Edge-disjoint paths in planar graphs , 1985, J. Comb. Theory, Ser. B.

[10]  Alan M. Frieze,et al.  Existence and Construction of Edge-Disjoint Paths on Expander Graphs , 1994, SIAM J. Comput..

[11]  Venkatesan Guruswami,et al.  Near-optimal hardness results and approximation algorithms for edge-disjoint paths and related problems , 2003, J. Comput. Syst. Sci..

[12]  Éva Tardos,et al.  Disjoint paths in densely embedded graphs , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[13]  Serge A. Plotkin.  Competitive Routing of Virtual Circuits in ATM Networks , 1995, IEEE J. Sel. Areas Commun..

[14]  Thomas Lengauer,et al.  Combinatorial algorithms for integrated circuit layout , 1990, Applicable theory in computer science.

[15]  Haruko Okamura,et al.  Multicommodity flows in planar graphs , 1981, J. Comb. Theory, Ser. B.

[16]  Haruko Okamura,et al.  Multicommodity flows in graphs , 1983, Discret. Appl. Math..

[17]  Christian Scheideler,et al.  Simple On-Line Algorithms for the Maximum Disjoint Paths Problem , 2004, Algorithmica.

[18]  Jon M. Kleinberg,et al.  Decision algorithms for unsplittable flow and the half-disjoint paths problem , 1998, STOC '98.

[19]  Robin Thomas,et al.  Quickly Excluding a Planar Graph , 1994, J. Comb. Theory, Ser. B.

[20]  Robin Thomas,et al.  Planar Separators , 1994, SIAM J. Discret. Math..

[21]  Carsten Thomassen,et al.  Highly Connected Sets and the Excluded Grid Theorem , 1999, J. Comb. Theory, Ser. B.

[22]  Paul D. Seymour,et al.  Graph minors. III. Planar tree-width , 1984, J. Comb. Theory B.

[23]  Sanjeev Khanna,et al.  The all-or-nothing multicommodity flow problem , 2004, STOC '04.

[24]  Éva Tardos,et al.  Approximations for the disjoint paths problem in high-diameter planar networks , 1995, STOC '95.

[25]  Mihalis Yannakakis,et al.  Primal-Dual Approximation Algorithms for Integral Flow and Multicut in Trees, with Applications to Matching and Set Cover , 1993, ICALP.

[26]  Paul D. Seymour,et al.  Graph Minors: XV. Giant Steps , 1996, J. Comb. Theory, Ser. B.

[27]  Prabhakar Raghavan,et al.  Randomized rounding: A technique for provably good algorithms and algorithmic proofs , 1985, Comb..

[28]  Sanjeev Khanna,et al.  Multicommodity flow, well-linked terminals, and routing problems , 2005, STOC '05.

[29]  Frank Thomson Leighton,et al.  Circuit Switching: A Multicommodity Flow Based Approach , 1995 .

[30]  Yuval Rabani,et al.  On-line admission control and circuit routing for high performance computing and communication , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[31]  N. Linial,et al.  Low distortion euclidean embeddings of trees , 1998 .