A Rounding by Sampling Approach to the Minimum Size k-Arc Connected Subgraph Problem

  • Bundit Laekhanukit
  • Shayan Oveis Gharan
  • Mohit Singh
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7391)


In the k-arc connected subgraph problem, we are given a directed graph G and an integer k and the goal is the find a subgraph of minimum cost such that there are at least k-arc disjoint paths between any pair of vertices. We give a simple (1 + 1/k)-approximation to the unweighted variant of the problem, where all arcs of G have the same cost. This improves on the 1 + 2/k approximation of Gabow et al. [GGTW09].

Similar to the 2-approximation algorithm for this problem [FJ81], our algorithm simply takes the union of a k in-arborescence and a k out-arborescence. The main difference is in the selection of the two arborescences. Here, inspired by the recent applications of the rounding by sampling method (see e.g. [AGM+10, MOS11, OSS11, AKS12]), we select the arborescences randomly by sampling from a distribution on unions of k arborescences that is defined based on an extreme point solution of the linear programming relaxation of the problem. In the analysis, we crucially utilize the sparsity property of the extreme point solution to upper-bound the size of the union of the sampled arborescences.

To complement the algorithm, we also show that the integrality gap of the minimum cost strongly connected subgraph problem (i.e., when k = 1) is at least 3/2 − ε, for any ε > 0. Our integrality gap instance is inspired by the integrality gap example of the asymmetric traveling salesman problem [CGK06], hence providing further evidence of connections between the approximability of the two problems.


Minimum Cost Linear Programming Relaxation Subgraph Problem Asymmetric Travel Salesman Problem Extreme Point Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AGM+10]
    Asadpour, A., Goemans, M.X., Madry, A., Gharan, S.O., Saberi, A.: An O(logn/ loglogn)-approximation algorithm for the asymmetric traveling salesman problem. In: SODA, pp. 379–389 (2010)Google Scholar
  2. [AKS12]
    An, H.-C., Kleinberg, R., Shmoys, D.B.: Improving Christofides’ algorithm for the s-t path tsp. In: STOC (to appear, 2012)Google Scholar
  3. [CGK06]
    Charikar, M., Goemans, M.X., Karloff, H.J.: On the integrality ratio for the asymmetric traveling salesman problem. Math. Oper. Res. 31(2), 245–252 (2006); Preliminary version in FOCS 2004Google Scholar
  4. [CT00]
    Cheriyan, J., Thurimella, R.: Approximating minimum-size k-connected spanning subgraphs via matching. SIAM J. Comput. 30(2), 528–560 (2000); Preliminary version in FOCS 1996Google Scholar
  5. [CV02]
    Carr, R.D., Vempala, S.: Randomized metarounding. Random Struct. Algorithms 20(3), 343–352 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  6. [Edm73]
    Edmonds, J.: Edge-disjoint branchings. In: Combinatorial algorithms (Courant Comput. Sci. Sympos. 9, New York Univ., New York, 1972), pp. 91–96. (1973)Google Scholar
  7. [FJ81]
    Frederickson, G.N., JáJá, J.: Approximation algorithms for several graph augmentation problems. SIAM J. Comput. 10(2), 270–283 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  8. [Fra79]
    Frank, A.: Covering branchings. Acta Scientiarum Mathematicarum (Szeged) 41, 77–81 (1979)zbMATHGoogle Scholar
  9. [Gab91]
    Gabow, H.N.: A matroid approach to finding edge connectivity and packing arborescences. In: STOC, pp. 112–122 (1991)Google Scholar
  10. [Gab04]
    Gabow, H.N.: Special edges, and approximating the smallest directed k-edge connected spanning subgraph. In: SODA, pp. 234–243 (2004)Google Scholar
  11. [GG08]
    Gabow, H.N., Gallagher, S.: Iterated rounding algorithms for the smallest k-edge connected spanning subgraph. In: SODA, pp. 550–559 (2008)Google Scholar
  12. [GGTW09]
    Gabow, H.N., Goemans, M.X., Tardos, É., Williamson, D.P.: Approximating the smallest k-edge connected spanning subgraph by LP-rounding. Networks 53(4), 345–357 (2009); Preliminary version in SODA 2005Google Scholar
  13. [HK70]
    Held, M., Karp, R.: The traveling salesman problem and minimum spanning trees. Operations Research 18, 1138–1162 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  14. [Jai01]
    Jain, K.: A factor 2 approximation algorithm for the generalized steiner network problem. Combinatorica 21(1), 39–60 (2001); Preliminary version in FOCS 1998Google Scholar
  15. [KRY94]
    Khuller, S., Raghavachari, B., Young, N.E.: Approximating the minimum equivalent digraph. In: Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 1994, pp. 177–186. Society for Industrial and Applied Mathematics, Philadelphia (1994)Google Scholar
  16. [KRY96]
    Khuller, S., Raghavachari, B., Young, N.E.: On strongly connected digraphs with bounded cycle length. Discrete Applied Mathematics 69(3), 281–289 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  17. [MOS11]
    Manshadi, V.H., Gharan, S.O., Saberi, A.: Online stochastic matching: Online actions based on offline statistics. In: SODA, pp. 1285–1294 (2011)Google Scholar
  18. [MT04]
    Melkonian, V., Tardos, E.: Algorithms for a network design problem with crossing supermodular demands. Networks 43(4), 256–265 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  19. [OSS11]
    Gharan, S.O., Saberi, A., Singh, M.: A randomized rounding approach to the traveling salesman problem. In: FOCS, pp. 550–559 (2011)Google Scholar
  20. [Sch03]
    Schrijver, A.: Combinatorial Optimization. Springer (2003)Google Scholar
  21. [Vet01]
    Vetta, A.: Approximating the minimum strongly connected subgraph via a matching lower bound. In: SODA, pp. 417–426 (2001)Google Scholar
  22. [ZNI03]
    Zhao, L., Nagamochi, H., Ibaraki, T.: A linear time 5/3-approximation for the minimum strongly-connected spanning subgraph problem. Inf. Process. Lett. 86, 63–70 (2003)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Bundit Laekhanukit
    • 1
  • Shayan Oveis Gharan
    • 2
  • Mohit Singh
    • 3
    • 4
  1. 1.School of Computer ScienceMcGill UniversityCanada
  2. 2.Department of Management Science and EngineeringStanford UniversityUSA
  3. 3.McGill UniversityCanada
  4. 4.Microsoft ResearchRedmondUSA

Personalised recommendations