## Abstract

In the undirected Edge-Disjoint Paths problem with Congestion (EDPwC), we are given an undirected graph with *V* nodes, a set of terminal pairs and an integer *c*. The objective is to route as many terminal pairs as possible, subject to the constraint that at most *c* demands can be routed through any edge in the graph. When *c* = 1, the problem is simply referred to as the Edge-Disjoint Paths (EDP) problem. In this paper, we study the hardness of EDPwC in undirected graphs.

Our main result is that for every *ɛ* > 0 there exists an *α* > 0 such that for 1 ⩽ *c* ⩽ \( \frac{{\alpha log log V }} {{\log \log \log V}} \), it is hard to distinguish between instances where we can route all terminal pairs on edge-disjoint paths, and instances where we can route at most a \( {1 \mathord{\left/ {\vphantom {1 {\left( {\log V} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\log V} \right)}}^{\frac{{1 - \varepsilon }} {{c + 2}}} \) fraction of the terminal pairs, even if we allow congestion *c*. This implies a \( \left( {\log V} \right)^{\frac{{1 - \varepsilon }} {{c + 2}}} \) hardness of approximation for EDPwC and an *Ω*(log log*V*/log log log*V*) hardness of approximation for the undirected congestion minimization problem. These results hold assuming NP ⊊ ∪_{
d
}ZPTIME(\( 2^{\log ^{d_n } } \)).

In the case that we do not require perfect completeness, i.e. we do not require that all terminal pairs are routed for “yes-instances”, we can obtain a slightly better inapproximability ratio of \( \left( {\log V} \right)^{\frac{{1 - \varepsilon }} {{c + 1}}} \). Note that by setting *c*=1 this implies that the regular EDP problem is \( \left( {\log V} \right)^{\frac{1} {2} - \varepsilon } \) hard to approximate.

Using standard reductions, our results extend to the node-disjoint versions of these problems as well as to the directed setting. We also show a \( \left( {\log V} \right)^{\frac{{1 - \varepsilon }} {{c + 1}}} \) inapproximability ratio for the All-or-Nothing Flow with Congestion (ANFwC) problem, a relaxation of EDPwC, in which the flow unit routed between the source-sink pairs does not have to follow a single path, so the resulting flow is not necessarily integral.

This is a preview of subscription content, access via your institution.

## References

- [1]
M. Andrews: Hardness of buy-at-bulk network design, in:

*Proceedings of the 45th Annual Symposium on Foundations of Computer Science*, pages 115–124, Rome, Italy, October 2004. - [2]
M. Andrews, J. Chuzhoy, S. Khanna and L. Zhang: Hardness of the undirected edge-disjoint paths problem with congestion, in:

*Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science*, pages 226–244, 2005. - [3]
M. Andrews and L. Zhang: Hardness of the undirected edge-disjoint paths problem, in:

*Proceedings of the 37th Annual ACM Symposium on Theory of Computing*, pages 276–283, 2005. - [4]
M. Andrews and L. Zhang: Hardness of the undirected congestion minimization problem, in:

*Proceedings of the 37th Annual ACM Symposium on Theory of Computing*, pages 284–293, 2005. - [5]
M. Andrews and L. Zhang: Hardness of the edge-disjoint paths problem with congestion, Manuscript, 2005. Available at: ]http://ect.bell-labs.com/who/dmandrews/publications/edp-congestion.ps.

- [6]
M. Andrews and L. Zhang: Logarithmic hardness of the directed congestion minimization problem, in:

*Proceedings of the 38th Annual ACM Symposium on Theory of Computing*, pages 517–526, 2006. - [7]
Y. Aumann and Y. Rabani: An

*O*(log*k*) approximate min-cut max-flow theorem and approximation algorithm,*SIAM Journal on Computing***27(1)**(1998), 291–301. - [8]
Y. Azar and O. Regev: Strongly polynomial algorithms for the unsplittable flow problem, in:

*Proceedings of the 8th Integer Programming and Combinatorial Optimization Conference*, pages 15–29, 2001. - [9]
A. Baveja and A. Srinivasan: Approximation algorithms for disjoint paths and related routing and packing problems,

*Mathematics of Operations Research***25(2)**(2000), 255–280. - [10]
C. Chekuri, S. Khanna and F. B. Shepherd: The all-or-nothing multicommodity flow problem, in:

*Proceedings of the 36th Annual ACM Symposium on Theory of Computing*, pages 156–165, 2004. - [11]
C. Chekuri, S. Khanna and F. B. Shepherd: Edge-disjoint paths in planar graphs with constant congestion, in:

*Proceedings of the 38th Annual ACM Symposium on Theory of Computing*, pages 757–766, 2006. - [12]
C. Chekuri, S. Khanna and F. B. Shepherd: Multicommodity flow, well-linked terminals, and routing problems; in:

*Proceedings of the 37th Annual ACM Symposium on Theory of Computing*, pages 183–192, 2005. - [13]
C. Chekuri, S. Khanna and F. B. Shepherd: An

*O*(\( \sqrt n \))-approximation and integrality gap for disjoint paths and UFP,*Theory of Computing***2**(2006), 137–146. - [14]
C. Chekuri and S. Khanna: Edge disjoint paths revisited, in:

*Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms*, pages 628–637, 2003. - [15]
C. Chekuri, M. Mydlarz and F. B. Shepherd: Multicommodity demand flow in a tree and packing integer programs,

*ACM Transactions on Algorithms***3(3)**(2007), Art.no. 27. - [16]
J. Chuzhoy and S. Khanna: New hardness results for undirected edge disjoint paths, Manuscript, 2005. Available at: http://ttic.uchicago.edu/~cjulia/papers/edpc.pdf.

- [17]
J. Chuzhoy, V. Guruswami, S. Khanna and K. Talwar: Hardness of routing with congestion in directed graphs, in:

*Proceedings of 39th Annual ACM Symposium on Theory of Computing*, pages 165–178, 2007. - [18]
L. Engebretsen: The nonapproximability of non-boolean predicates,

*SIAM Journal on Discrete Mathematics***18**(2004), 114–129. - [19]
P. Erdős and H. Sachs: Reguläre Graphen gegebener Taillenweite mit minimaler Knotenzahl,

*Wiss. Z. Uni. Halle-Wittenburg (Math. Nat.)***12**(1963), 251–257. - [20]
A. Frank: Edge-disjoint paths in planar graphs,

*J. of Combinatorial Theory, Ser. B***2**(1985), 164–178. - [21]
S. Fortune, J. Hopcroft and J. Wyllie: The directed subgraph homeomorphism problem,

*Theoretical Computer Science***10(2)**(1980), 111–121. - [22]
Z. Friggstad and M. R. Salavatipour: Approximability of packing disjoint cycles, in:

*Proceedings of 18th International Symposium on Algorithms and Computation*, pages 304–315, 2007. - [23]
M. R. Garey and D. S. Johnson:

*Computers and intractability: A guide to the theory of NP-completeness*; Freeman, 1979. - [24]
N. Garg, V. Vazirani and M. Yannakakis: Primal-dual approximation algorithms for integral flow and multicut in trees,

*Algorithmica***18(1)**(1997), 3–20. - [25]
V. Guruswami, S. Khanna, R. Rajaraman, F. B. Shepherd and M. Yannakakis: Near-optimal hardness results and approximation algorithms for edge-disjoint paths and related problems,

*J. Comput. Syst. Sci.***67(3)**(2003), 473–496. - [26]
V. Guruswami and K. Talwar: Hardness of low congestion routing in undirected graphs,

*Manuscript*, 2005. - [27]
J. Håstad and S. Khot: Query efficient PCPs with perfect completeness,

*Theory of Computing***1(7)**(2005), 119–148. - [28]
J. Håstad and A. Wigderson: Simple analysis of graph tests for linearity and PCP,

*Random Structures and Algorithms***22(2)**(2003), 139–160. - [29]
J. M. Kleinberg and É. Tardos: Approximations for the disjoint paths problem in high-diameter planar networks,

*Journal of Computer and System Sciences***57**(1998), 61–73. - [30]
J. M. Kleinberg and É. Tardos: Disjoint paths in densely embedded graphs, in:

*Proceedings of the 36th Annual Symposium on Foundations of Computer Science*, page 52, 1995. - [31]
J. M. Kleinberg: Approximation algorithms for disjoint paths problems, PhD thesis, MIT, Cambridge, MA, 1996.

- [32]
S. G. Kolliopoulos and C. Stein: Approximating disjoint-path problems using greedy algorithms and packing integer programs, in:

*Proceedings of the Conference on Integer Programming and Combinatorial Approximation*, pages 153–168, 1998. - [33]
M. Krivelevich, Z. Nutov and R. Yuster: Approximation algorithms for cycle packing problems, in:

*Proceedings of the 16th ACM-SIAM Symposium on Discrete Algorithms*, pages 556–561, 2005. - [34]
P. Raghavan and C. D. Thompson: Randomized rounding: A technique for provably good algorithms and algorithmic proofs;

*Combinatorica***7(4)**(1987), 365–374. - [35]
R. Raz: A parallel repetition theorem,

*SIAM Journal on Computing***27(3)**(1998), 763–803. - [36]
N. Robertson and P. D. Seymour: An outline of a disjoint paths algorithm, in:

*Paths, Flows and VLSI-Layout (B. Korte, L. Lovász, H. J. Prömel and A. Schrijver*,*eds.)*, pages 267–292, Springer-Verlag, Berlin, 1990. - [37]
A. Samorodnitsky and L. Trevisan: A PCP characterization of NP with optimal amortized query complexity, in:

*Proceedings of the 32nd Annual ACM Symposium on Theory of Computing*, pages 191–199, 2000. - [38]
A. Srinivasan: Improved approximations for edge-disjoint paths, unsplittable flow, and related routing problems; in:

*Proceedings of the 38th Symposium on Foundations of Computer Science*, pages 416–425, 1997. - [39]
K. Varadarajan and G. Venkataraman: Graph decomposition and a greedy algorithm for edge-disjoint paths, in:

*Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms*, pages 379–380, 2004.

## Author information

### Affiliations

### Corresponding author

## Additional information

This work was done when the author was at the Dept. of Computer Science and Engineering, University of Washington. Research supported in part by NSF CCF-0343672, and Sloan and Packard fellowships.

Supported in part by NSF Career Award CCR-0093117 and by NSF Award CCF-0635084.

## Rights and permissions

## About this article

### Cite this article

Andrews, M., Chuzhoy, J., Guruswami, V. *et al.* Inapproximability of Edge-Disjoint Paths and low congestion routing on undirected graphs.
*Combinatorica* **30, **485–520 (2010). https://doi.org/10.1007/s00493-010-2455-9

Received:

Published:

Issue Date:

### Mathematics Subject Classification (2000)

- 68Q17
- 05C38