Network design for vertex connectivity
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Approximating Minimum-Power k-Connectivity
ADHOC-NOW '08 Proceedings of the 7th international conference on Ad-hoc, Mobile and Wireless Networks
On k-connectivity problems with sharpened triangle inequality
Journal of Discrete Algorithms
An almost O(log k)-approximation for k-connected subgraphs
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Approximating minimum-power edge-covers and 2,3-connectivity
Discrete Applied Mathematics
Approximating Node-Connectivity Augmentation Problems
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
On minimum power connectivity problems
ESA'07 Proceedings of the 15th annual European conference on Algorithms
Approximating minimum-power degree and connectivity problems
LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
Approximating survivable networks with minimum number of Steiner points
WAOA'10 Proceedings of the 8th international conference on Approximation and online algorithms
An improved approximation algorithm for minimum-cost subset k-connectivity
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Approximating minimum power covers of intersecting families and directed connectivity problems
APPROX'06/RANDOM'06 Proceedings of the 9th international conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th international conference on Randomization and Computation
Approximating subset k-connectivity problems
WAOA'11 Proceedings of the 9th international conference on Approximation and Online Algorithms
Approximating subset k-connectivity problems
Journal of Discrete Algorithms
Survivable network activation problems
Theoretical Computer Science
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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 $O(\min \{k,\frac{n}{\sqrt{n-k}}\})$ for both directed and undirected graphs, and $O(\ln k)$ for undirected graphs with $n \geq 6k^2$, where $n$ is the number of nodes in the input graph. Our first algorithm has approximation ratio $O(\frac{n}{n-k}\ln^2 k)$, which is $O(\ln^2 k)$ except for very large values of $k$, namely, $k=n-o(n)$. This algorithm is based on a new result on $\ell$-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 $O(\sqrt{n} \ln k)$ for all values of $n,k$. Combining these two gives an algorithm with approximation ratio $O(\ln k \cdot \min \{\sqrt{k},\frac{n}{n-k} \ln k\})$, which asymptotically improves the best known approximation guarantee for directed graphs for all values of $n,k$, and for undirected graphs for $k\sqrt{n/6}$. Moreover, this is the first algorithm that has an approximation guarantee better than $\Theta(k)$ for all values of $n,k$. Our approximation ratio also provides an upper bound on the integrality gap of the standard LP-relaxation.