Improved LP-rounding Approximations for the k-Disjoint Restricted Shortest Paths Problem

  • Longkun Guo
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8497)


Let G = (V, E) be a given (directed) graph in which every edge is with a cost and a delay that are nonnegative. The k-disjoint restricted shortest path (kRSP) problem is to compute k (edge) disjoint minimum cost paths between two distinct vertices s, t ∈ V, such that the total delay of these paths are bounded by a given delay constraint \(D\in\mathbb{R}_{0}^{+}\). This problem is known to be NP-hard, even when k = 1 [4]. Approximation algorithms with bifactor ratio \((1+\frac{1}{r},\, r(1+\frac{2(\log r+1)}{r})(1+\epsilon))\) and \((1+\frac{1}{r},\, r(1+\frac{2(\log r+1)}{r}))\) have been developed for its special case when k = 2 respectively in [11] and [3]. For general k, an approximation algorithm with ratio (1, O(ln n)) has been developed for a weaker version of kRSP, the k bi-constraint path problem of computing k disjoint st-paths to satisfy the given cost constraint and delay constraint simultaneously [7].

In this paper, an approximation algorithm with bifactor ratio (2, 2) is first given for the kRSP problem. Then it is improved such that for any resulted solution, there exists a real number 0 ≤ α ≤ 2 that the delay and the cost of the solution is bounded, respectively, by α times and 2 − α times of that of an optimal solution. These two algorithms are both based on rounding a basic optimal solution of a LP formula, which is a relaxation of an integral linear programming (ILP) formula for the kRSP problem. The key observation of the two ratio proofs is to show that, the fractional edges of a basic solution to the LP formula will compose a graph in which the degree of every vertex is exactly 2. To the best of our knowledge, it is the first algorithm with a single factor polylogarithmic ratio for the kRSP problem.


LP rounding flow theory k-disjoint restricted shortest path problem bifactor approximation algorithm 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network flows: theory, algorithms, and applications (1993)Google Scholar
  2. 2.
    Bhatia, R., Kodialam, M.: TV Lakshman. Finding disjoint paths with related path costs. Journal of Combinatorial Optimization 12(1), 83–96 (2006)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Chao, P., Hong, S.: A new approximation algorithm for computing 2-restricted disjoint paths. IEICE Transactions on Information and Systems 90(2), 465–472 (2007)Google Scholar
  4. 4.
    Garey, M.R., Johnson, D.S.: Computers and intractability. Freeman, San Francisco (1979)zbMATHGoogle Scholar
  5. 5.
    Guo, L., Shen, H.: On the complexity of the edge-disjoint min-min problem in planar digraphs. Theoretical Computer Science 432, 58–63 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Guo, L., Shen, H.: On finding min-min disjoint paths. Algorithmica 66(3), 641–653 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Guo, L., Shen, H., Liao, K.: Improved approximation algorithms for computing k disjoint paths subject to two constraints. In: Du, D.-Z., Zhang, G. (eds.) COCOON 2013. LNCS, vol. 7936, pp. 325–336. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  8. 8.
    Li, C.L., McCormick, T.S., Simich-Levi, D.: The complexity of finding two disjoint paths with min-max objective function. Discrete Applied Mathematics 26(1), 105–115 (1989)CrossRefGoogle Scholar
  9. 9.
    Lorenz, D.H., Raz, D.: A simple efficient approximation scheme for the restricted shortest path problem. Operations Research Letters 28(5), 213–219 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Misra, S., Xue, G., Yang, D.: Polynomial time approximations for multi-path routing with bandwidth and delay constraints. In: INFOCOM 2009, pp. 558–566. IEEE (2009)Google Scholar
  11. 11.
    Orda, A., Sprintson, A.: Efficient algorithms for computing disjoint QoS paths. In: INFOCOM 2004, vol. 1, pp. 727–738. IEEE (2004)Google Scholar
  12. 12.
    Schrijver, A.: Theory of linear and integer programming. John Wiley & Sons Inc. (1998)Google Scholar
  13. 13.
    Suurballe, J.W.: Disjoint paths in a network. Networks 4(2) (1974)Google Scholar
  14. 14.
    Suurballe, J.W., Tarjan, R.E.: A quick method for finding shortest pairs of disjoint paths. Networks 14(2) (1984)Google Scholar
  15. 15.
    Xu, D., Chen, Y., Xiong, Y., Qiao, C., He, X.: On the complexity of and algorithms for finding the shortest path with a disjoint counterpart. IEEE/ACM Transactions on Networking 14(1), 147–158 (2006)CrossRefGoogle Scholar
  16. 16.
    Xue, G., Zhang, W., Tang, J., Thulasiraman, K.: Polynomial time approximation algorithms for multi-constrained qos routing. IEEE/ACM Transactions on Networking 16(3), 656–669 (2008)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Longkun Guo
    • 1
  1. 1.College of Mathematics and Computer ScienceFuzhou UniversityChina

Personalised recommendations