# Approximation algorithms for graph augmentation

## Abstract

We study the problem of increasing the connectivity^{1} of a graph at an optimal cost. Since the general problem is *NP*-hard, we focus on efficient approximation schemes that come within a constant factor from the optimal. Previous algorithms either do not take edge costs into consideration, or run slower than our algorithm. Our algorithm takes as input an undirected graph G_{0} = (*V*, E_{0}) on *n* vertices, that is not necessarily connected, and a set *Feasible* of m weighted edges on *V*, and outputs a subset *Aug* of edges which when added to G_{0} make it 2-connected. The weight of *Aug*, when G_{0} is initially connected, is no more than twice the weight of the least weight subset of edges of *Feasible* that increases the connectivity to 2. The running time of our algorithm is *O(m + n* logn). We also study the problem of increasing the edge connectivity of any graph *G*, to *k*, within a factor of 2 (for any *k* > 0). The running time of this algorithm is *O(nk* log n(*m + n* log *n*)). We observe that when *k* is *odd* we can use different techniques to obtain an approximation factor of 2 for increasing edge connectivity from *k* to (*k*+1) in *O(kn*^{2}) time.

## Preview

Unable to display preview. Download preview PDF.

## References

- [AHU]A.V. Aho, J.E. Hopcroft and J.D. Ullman,
*The design and analysis of computer algorithms*, Addison-Wesley, 1974.Google Scholar - [DKL76]E.A. Dinits, A.V. Karzanov and M.L. Lomosonov, “On the structure of a family of minimal weighted cuts in a graph,”
*Studies in Discrete Optimization*[In Russian], A.A. Fridman (ear decomposition), Nauka, Moscow, pp. 290–306, 1976.Google Scholar - [NGM90]D. Naor, D. Gusfield and C. Martel, “A fast algorithm for optimally increasing the edge-connectivity,”
*31st Annual Symposium on Foundations of Computer Science*, pp. 698–707, (1990).Google Scholar - [Ed79]J. Edmonds, “Matroid intersection,”
*Annals of Discrete Mathematics*, No. 4, pp. 185–204, (1979).Google Scholar - [Ev79]S. Even,
*Graph Algorithms*, Computer Science Press, Potomac, Md., 1979.Google Scholar - [ET76]K. P. Eswaran and R. E. Tarjan, “Augmentation problems,”
*SIAM Journal on Computing*, Vol. 5, No. 4, pp. 653–665, (1976).CrossRefGoogle Scholar - [Fr90]A. Frank, “Augmenting graphs to meet edge-connectivity requirements,”
*31st Annual Symposium on Foundations of Computer Science*, pp. 708–718, (1990).Google Scholar - [FJ81]G. N. Frederickson and J. JáJá, “Approximation algorithms for several graph augmentation problems,”
*SIAM Journal on Computing*, Vol. 10, No. 2, pp. 270–283, (1981).CrossRefGoogle Scholar - [FJ82]G. N. Frederickson and J. JáJá, On the relationship between the biconnectivity augmentation and traveling salesman problems,”
*Theoretical Computer Science*, Vol. 19, No. 2, pp. 189–201, (1982).CrossRefGoogle Scholar - [FT89]A. Frank and E. Tardos, “An application of submodular flows,”
*Linear Algebra and its Applications*, 114/115, pp. 320–348, (1989).Google Scholar - [G91a]H. N. Gabow, “A matroid approach to finding edge connectivity and packing arborescences,”
*23rd Annual Symposium on Theory of Computing*, pp. 112–122, (1991).Google Scholar - [G91b]H. N. Gabow, “Applications of a poset representation to edge connectivity and graph rigidity,”
*32nd Annual Symposium on Foundations of Computer Science*, pp. 812–822, (1991).Google Scholar - [GGST86]H. N. Gabow, Z. Galil, T. Spencer and R. E. Tarjan, “Efficient algorithms for finding minimum spanning trees in undirected and directed graphs,”
*Combinatorica*, 6 (2), pp. 109–122, (1986).Google Scholar - [HR91a]T. S. Hsu and V. Ramachandran, “A linear time algorithm for triconnectivity augmentation,”
*32nd Annual Symposium on Foundations of Computer Science*, pp. 548–559, (1991).Google Scholar - [HR91b]T. S. Hsu and V. Ramachandran, “On finding a smallest augmentation to biconnect a graph,”
*2nd*Annual International Symposium on Algorithms, Springer Verlag LNCS 557, pp. 326–335, (1991).Google Scholar - [HT84]D. Harel and R. E. Tarjan, “Fast algorithms for finding nearest common ancestors,”
*SIAM Journal on Computing*, Vol 13, No. 2, pp. 338–355, (1984).CrossRefGoogle Scholar - [KB91]G. Kant and H. Bodlaender, “Planar Graph Augmentation Problems,”
*1991*Workshop on Algorithms and Data Structures, pp. 286–298, (1991).Google Scholar - [KT86]A.V. Karzanov and E.A, Timofeev, “Efficient algorithm for finding all minimal edge cuts of a nonoriented graph,”
*Cybernetics*, pp. 156–162, Translated from*Kibernetika*, No. 2, pp. 8–12, (1986).Google Scholar - [KV92]S. Khuller and U. Vishkin, “Biconnectivity approximations and graph carvings,” Technical Report UMIACS-TR-92-5, CS-TR-2825, Univ. of Maryland, January (1992), Also to appear in
*24th Annual Symposium on Theory of Computing*, (1992).Google Scholar - [NGM90]D. Naor, D. Gusfield and C. Martel, “A fast algorithm for optimally increasing the edge-connectivity,”
*31st Annual Symposium on Foundations of Computer Science*, pp. 698–707, (1990).Google Scholar - [RG77]A. Rosenthal and A. Goldner, “Smallest augmentations to biconnect a graph,”
*SIAM Journal on Computing*, Vol. 6, No. 1, pp. 55–66, (1977).CrossRefGoogle Scholar - [Wa88]T. Watanabe, “An efficient augmentation to k-edge connect a graph,” Tech. Report C-23, Department of Applied Math., Hiroshima University, April 1988.Google Scholar
- [WN87]T. Watanabe and A. Nakamura, “Edge-connectivity augmentation problems,”
*Journal of Comp. and Sys. Sciences*, 35 (1), pp. 96–144, (1987).CrossRefGoogle Scholar