Approximation Algorithms for Edge-Disjoint Paths and Unsplittable Flow

In the maximum edge-disjoint paths problem (MEDP) the input consists of a graph and a set of requests (pairs of vertices), and the goal is to connect as many requests as possible along edge-disjoint paths. We give a survey of known results about the complexity and approximability of MEDP and sketch some of the main ideas that have been used to obtain approximation algorithms for the problem. We consider also the generalization of MEDP where the edges of the graph have capacities and each request has a profit and a demand, called the unsplittable flow problem.

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

[2]  Adam Tauman Kalai,et al.  Admission Control to Minimize Rejections , 2003, Internet Math..

[3]  George L. Nemhauser,et al.  Note--On "Location of Bank Accounts to Optimize Float: An Analytic Study of Exact and Approximate Algorithms" , 1979 .

[4]  Johan Håstad,et al.  Clique is hard to approximate within n/sup 1-/spl epsiv// , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[5]  Gerhard J. Woeginger,et al.  Online Algorithms , 1998, Lecture Notes in Computer Science.

[6]  Hisao Tamaki,et al.  Routing a Permutation in the Hypercube by Two Sets of Edge Disjoint Paths , 1997, J. Parallel Distributed Comput..

[7]  Alan M. Frieze,et al.  Optimal construction of edge-disjoint paths in random graphs , 1994, SODA '94.

[8]  Aravind Srinivasan,et al.  Approximation Algorithms for Disjoint Paths and Related Routing and Packing Problems , 2000, Math. Oper. Res..

[9]  Sanjeev Khanna,et al.  Edge disjoint paths revisited , 2003, SODA '03.

[10]  Christian Scheideler,et al.  Simple on-line algorithms for the maximum disjoint paths problem , 2001, SPAA '01.

[11]  Thomas Erlebach,et al.  Approximation Algorithms and Complexity Results for Path Problems in Trees of Rings , 2001, MFCS.

[12]  John E. Hopcroft,et al.  The Directed Subgraph Homeomorphism Problem , 1978, Theor. Comput. Sci..

[13]  Chandra Chekuri,et al.  Multicommodity Demand Flow in a Tree , 2003, ICALP.

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

[15]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[16]  Joseph Y.-T. Leung,et al.  Efficient algorithms for interval graphs and circular-arc graphs , 1982, Networks.

[17]  Mihalis Yannakakis,et al.  Primal-dual approximation algorithms for integral flow and multicut in trees , 1997, Algorithmica.

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

[19]  Klaus Jansen,et al.  The Maximum Edge-Disjoint Paths Problem in Bidirected Trees , 2001, SIAM J. Discret. Math..

[20]  Klaus Jansen,et al.  Approximation Algorithms for Combinatorial Optimization , 2000 .

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

[22]  Robin Milner.  An Action Structure for Synchronous pi-Calculus , 1993, FCT.

[23]  Aravind Srinivasan,et al.  Improved approximations for edge-disjoint paths, unsplittable flow, and related routing problems , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[24]  Takao Nishizeki,et al.  Finding Edge-Disjoint Paths in Partial k -Trees , 2000, Algorithmica.

[25]  Amit Kumar,et al.  Approximation Algorithms for the Unsplittable Flow Problem , 2002, Algorithmica.

[26]  Clifford Stein,et al.  Approximating Disjoint-Path Problems Using Greedy Algorithms and Packing Integer Programs ( Extended Abstract ) , 1998 .

[27]  Alan M. Frieze.  Edge-disjoint paths in expander graphs , 2000, SODA '00.

[28]  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.

[29]  Martin Skutella.  Approximating the single source unsplittable min-cost flow problem , 2002, Math. Program..

[30]  G. Nemhauser,et al.  Exceptional Paper—Location of Bank Accounts to Optimize Float: An Analytic Study of Exact and Approximate Algorithms , 1977 .

[31]  Christian Scheideler,et al.  Improved bounds for the unsplittable flow problem , 2002, SODA '02.

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

[33]  Michel X. Goemans,et al.  On the Single-Source Unsplittable Flow Problem , 1999, Comb..

[34]  Stefano Leonardi,et al.  On-line Network Routing , 1996, Online Algorithms.

[35]  Alexander Schrijver,et al.  Paths, Flows, and VLSI-Layout , 1990 .

[36]  P. Nguyen,et al.  On the Disjoint Paths , 1998 .

[37]  Thomas Erlebach,et al.  New Results for Path Problems in Generalized Stars, Complete Graphs, and Brick Wall Graphs , 2001, FCT.

[38]  Venkatesan Guruswami,et al.  Near-optimal hardness results and approximation algorithms for edge-disjoint paths and related problems , 1999, STOC '99.

[39]  Yossi Azar,et al.  Strongly Polynomial Algorithms for the Unsplittable Flow Problem , 2001, IPCO.

[40]  Peng-Jun Wan,et al.  Maximal throughput in wavelength-routed optical networks , 1998, Multichannel Optical Networks: Theory and Practice.

[41]  Aris Pagourtzis,et al.  Minimizing request blocking in all-optical rings , 2003, IEEE INFOCOM 2003. Twenty-second Annual Joint Conference of the IEEE Computer and Communications Societies (IEEE Cat. No.03CH37428).

[42]  Bin Ma,et al.  On the Inapproximability of Disjoint Paths and Minimum Steiner Forest with Bandwidth Constraints , 2000, J. Comput. Syst. Sci..

[43]  Peter Bro Miltersen,et al.  On Pseudorandom Generators in NC , 2001, MFCS.

[44]  J. Håstad.  Clique is hard to approximate withinn1−ε , 1999 .

[45]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.

[46]  Allan Borodin,et al.  Online computation and competitive analysis , 1998 .

[47]  Jon M. Kleinberg,et al.  Approximation algorithms for disjoint paths problems , 1996 .

[48]  Robin Milner,et al.  On Observing Nondeterminism and Concurrency , 1980, ICALP.

[49]  Hans L. Bodlaender,et al.  A Partial k-Arboretum of Graphs with Bounded Treewidth , 1998, Theor. Comput. Sci..

[50]  Alan M. Frieze,et al.  Optimal construction of edge-disjoint paths in random regular graphs , 2000, SODA '99.

[51]  Neil Robertson,et al.  Graph Minors .XIII. The Disjoint Paths Problem , 1995, J. Comb. Theory B.

[52]  Alan M. Frieze,et al.  Arc-disjoint paths in expander digraphs , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[53]  Petr Kolman,et al.  A note on the greedy algorithm for the unsplittable flow problem , 2003, Inf. Process. Lett..

[54]  Thomas Erlebach.  Scheduling connections in fast networks , 1999 .

[55]  Yoshio Okamoto,et al.  Greedy Edge-Disjoint Paths in Complete Graphs , 2003, WG.

[56]  Hans L. Bodlaender,et al.  A Tourist Guide through Treewidth , 1993, Acta Cybern..

[57]  Niklaus Wirth,et al.  Algorithms and Data Structures , 1989, Lecture Notes in Computer Science.

[58]  Christian Scheideler,et al.  Universal Routing Strategies for Interconnection Networks , 1998, Lecture Notes in Computer Science.

[59]  MATTHIAS MIDDENDORF,et al.  On the complexity of the disjoint paths problem , 1993, Comb..

[60]  J. Håstad.  Clique is hard to approximate within n 1-C , 1996 .

[61]  Ganesh Venkataraman,et al.  Graph decomposition and a greedy algorithm for edge-disjoint paths , 2004, SODA '04.

[62]  Frank Thomson Leighton,et al.  Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms , 1999, JACM.

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

[64]  Johan Håstad,et al.  Clique is hard to approximate within n1-epsilon , 1996, Electron. Colloquium Comput. Complex..

[65]  Russ Bubley,et al.  Randomized algorithms , 1995, CSUR.