# A 2-Approximation for Minimum Cost {0, 1, 2} Vertex Connectivity

• Lisa Fleischer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2081)

## Abstract

In survivable network design, each pair (i, j) of vertices is assigned a level of importance r ij. The vertex connectivity problem is to design a minimum cost network such that between each pair of vertices with importance level r, there are r vertex disjoint paths. There is no approximation algorithm known for this general problem. In this paper, we give a 2-approximation for the problem when r ∈ {0, 1, 2}V xV, improving on a previous known 3-approximation. This matches the best known approximation for the easier problem that requires that the paths be only edge-disjoint.

Our algorithm extends an iterative rounding algorithm that gives a 2-approximation for the edge-connectivity problem, for arbitrary connectivity requirements r. (K. Jain, A factor 2 approximation for the gen- eralized Steiner network problem.) This algorithm relies on well-known uncrossing lemma for tight edge cutsets. Our extension uses a new type of uncrossing lemma for tight cutsets that may include vertices as well as edges.

For r ∈ {1, k}V xV, k ≥ 3, we show that a) uncrossing tight cutsets is not possible, and b) any analysis for iterative rounding that depends directly on the largest fractional value in the linear programming solution cannot provide approximation guarantees better than the maximum connectivity requirement.

## Keywords

Approximation Algorithm Network Design Problem Basic Feasible Solution Vertex Connectivity Connectivity Problem
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.

## References

1. 1.
A. Bouchet. Greedy algorithm and symmetric matroids. Math. Programming, 38:147–159, 1987.
2. 2.
R. Chandrasekaran and S. N. Kabadi. Pseudomatroids. Discrete Math., 71: 205–217, 1988.
3. 3.
J. Cheriyan, T. Jordán, and Z. Nutov. Approximating k-outconnected subgraph problems. In Approximation algorithms for combinatorial optimization (Aarlborg), number 1444 in Lecture Notes in Comput. Sci., pages 77–88. Springer, Berlin, 1998.
4. 4.
R. Diestel. Graph Theory. Number 173 in Graduate Texts in Mathematics. Springer-Verlag, New York, 2nd edition, 2000.Google Scholar
5. 5.
M. X. Goemans, A. V. Goldberg, S. Plotkin, D. Shmoys, É. Tardos, and D. P. Williamson. Improved approximation algorithms for network design problems. In Proc. 5th Annual ACM-SIAM Symp. on Discrete Algorithms, pages 223–232, 1994.Google Scholar
6. 6.
M. X. Goemans and D. P. Williamson. The primal-dual method for approximation algorithms and its application to network design problems. In Hochbaum[9], pages 144–191.Google Scholar
7. 7.
M. Grötschel, L. Lovász, and A. Schrijver. Geometric Algorithms and Combinatorial Optimization. Springer-Verlag, 1988.Google Scholar
8. 8.
M. Grötschel, C. L. Monma, and M. Stoer. Computational results with a cutting plane algorithm for designing communication networks with low-connectivity constraints. Operations Research, 40(2):309–330, March-April 1992.
9. 9.
D. S. Hochbaum, editor. Approximation Algorithms for NP-Hard Problems. PWS Publishing Company, Boston, 1997.Google Scholar
10. 10.
K. Jain. A factor 2 approximation algorithm for the generalized Steiner network problem. In 39th Annual IEEE Symposium on Foundations of Computer Science, 1998.Google Scholar
11. 11.
S. Khuller. Approximation algorithms for finding highly connected subgraphs. In Hochbaum[9], pages 236-265.Google Scholar
12. 12.
S. Khuller and B. Raghavachari. Improved approximation algorithms for uniform connectivity problems. J. Algorithms, 1996.Google Scholar
13. 13.
G. Kortsarz and Z. Nutov. Approximating node connectivity problems via set covers. In Approximation Algorithms for Combinatorial Optimization (Proc. of APPROX 2000), number 1913 in Lecture Notes in Comp. Sci., pages 194–205. Springer-Verlag, 2000.Google Scholar
14. 14.
V. Melkonian and E. Tardos. Approximation algorithms for a directed network design problem. In 7th International Integer Programming and Combinatorial Optimization Conference, pages 345–360, 1999.Google Scholar
15. 15.
C. L. Monma and D. F. Shallcross. Methods for designing communications networks with certain two-connected survivability constraints. Operations Research, 1989.Google Scholar
16. 16.
R. Ravi and D. P. Williamson. An approximation algorithm for minimum-cost vertex-connectivity problems. Algorithmica, 18(1):21–43, 1997.
17. 17.
R. Ravi and D. P. Williamson, November 2000. Personal communication.Google Scholar