On the optimal vertex-connectivity augmentation
Journal of Combinatorial Theory Series B
Minimal edge-coverings of pairs of sets
Journal of Combinatorial Theory Series B
Improved approximation algorithms for uniform connectivity problems
Journal of Algorithms
On the existence of (k,l)-critical graphs
Discrete Mathematics
A representation for crossing set families with applications to submodular flow problems
SODA '93 Proceedings of the fourth annual ACM-SIAM Symposium on Discrete algorithms
Approximation algorithms for minimum-cost k-vertex connected subgraphs
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Approximating Minimum-Size k-Connected Spanning Subgraphs via Matching
SIAM Journal on Computing
On k-con-critically n-connected graphs
Journal of Combinatorial Theory Series B
High Connectivity Keeping Sets In n-Connected Graphs
Combinatorica
Approximation algorithms for network design with metric costs
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
An o(log2 k)-approximation algorithm for the k-vertex connected spanning subgraph problem
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Approximation schemes for minimum 2-connected spanning subgraphs in weighted planar graphs
ESA'05 Proceedings of the 13th annual European conference on Algorithms
Power optimization for connectivity problems
IPCO'05 Proceedings of the 11th international conference on Integer Programming and Combinatorial Optimization
On the L ∞ -norm of extreme points for crossing supermodular directed network LPs
IPCO'05 Proceedings of the 11th international conference on Integer Programming and Combinatorial Optimization
Tight approximation algorithm for connectivity augmentation problems
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
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We present two new approximation algorithms for the problem of finding a k-node connected spanning subgraph (directed or undirected) of minimum cost. The best known approximation guarantees for this problem were O(min (k,n/√n-k)) for both directed and undirected graphs, and O(ln k) for undirected graphs with n ≥ 6k2, where n is the number of nodes in the input graph. Our first algorithm has approximation ratio O(k/n-kln 2 k, which is O(ln2 k) except for very large values of k, namely, k=n-o(n). This algorithm is based on a new result on l-connected p-critical graphs, which is of independent interest in the context of graph theory. Our second algorithm uses the primal-dual method and has approximation ratio O(√n ln k) for all values of n,k. Combining these two gives an algorithm with approximation ratio O(ln k • min (√k, k/n-k ln k)), which asymptotically improves the best known approximation guarantee for directed graphs for all values of n,k, and for undirected graphs for k √n⁄6. Moreover, this is the first algorithm that has an approximation guarantee better than Θ(k) for all values of n,k. Our approximation ratio also provides an upper bound on the integrality gap of the standard LP-relaxation to the problem.As a byproduct, we also get the following result which is of independent interest. To get a faster implementation of our algorithms, we consider the problem of adding a minimum-cost edge set to increase the outconnectivity of a directed graph by Δ a graph is said to be l-outconnected from its node r if it contains l internally disjoint paths from r to any other node. The best known time complexity for the later problem is O(m3). For the particular case of Δ=1, we give a primal-dual algorithm with running time O(m2).