On the structure of minimum-weight k-connected spanning networks
SIAM Journal on Discrete Mathematics
Euclidean spanners: short, thin, and lanky
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
Approximation algorithms for NP-hard problems
Approximation algorithms for NP-hard problems
The primal-dual method for approximation algorithms and its application to network design problems
Approximation algorithms for NP-hard problems
Approximation algorithms for finding highly connected subgraphs
Approximation algorithms for NP-hard problems
Approximating geometrical graphs via “spanners” and “banyans”
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems
Journal of the ACM (JACM)
On approximability of the minimum-cost k-connected spanning subgraph problem
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Improved Greedy Algorithms for Constructing Sparse Geometric Spanners
SWAT '00 Proceedings of the 7th Scandinavian Workshop on Algorithm Theory
A Polynomial Time Approximation Scheme for Euclidean Minimum Cost k-Connectivity
ICALP '98 Proceedings of the 25th International Colloquium on Automata, Languages and Programming
Polynomial-Time Approximation Schemes for the Euclidean Survivable Network Design Problem
ICALP '02 Proceedings of the 29th International Colloquium on Automata, Languages and Programming
Experimental study of geometric t-spanners
Journal of Experimental Algorithmics (JEA)
Approximation Algorithms for Buy-at-Bulk Geometric Network Design
WADS '09 Proceedings of the 11th International Symposium on Algorithms and Data Structures
Approximation schemes for minimum 2-connected spanning subgraphs in weighted planar graphs
ESA'05 Proceedings of the 13th annual European conference on Algorithms
Experimental study of geometric t-spanners
ESA'05 Proceedings of the 13th annual European conference on Algorithms
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We present new polynomial-time approximation schemes (PTAS) for several basic minimum-cost multi-connectivity problems in geometrical graphs. We focus on low connectivity requirements. Each of our schemes either significantly improves the previously known upper time-bound or is the first PTAS for the considered problem. We provide a randomized approximation scheme for finding a biconnected graph spanning a set of points in a multi-dimensional Euclidean space and having the expected total cost within (1+Ɛ) of the optimum. For any constant dimension and Ɛ, our scheme runs in time O(n log n). It can be turned into Las Vegas one without affecting its asymptotic time complexity, and also efficiently derandomized. The only previously known truly polynomial-time approximation (randomized) scheme for this problem runs in expected time n ċ (log n)O((log log n)9) in the simples planer case. The efficiency of our scheme relies on transformations of nearly optimal low cost special spanners into sub-multigraphs having good decomposition and approximation properties and a simple subgraph connectivity characterization. By using merely the spanner transformations, we obtain a very fast polynomial-time approximation scheme for finding a minimum-cost k-edge connected multigraph spanning a set of points in a multi-dimensional O(n log n). Furthermore, by showing a low-cost transformation of a k-edge connected graph maintaining the k-edge connectivity and developing novel decomposition properties, we derive a PTAS for Euclidean minimum-cost k-edge connectivity. It is substantially faster than that previously known. Finally, by extending our techiques, we obtain the first PTAS for the problem of Euclidean minimum-cost Steiner biconnectivity. This scheme runs in time O(n log n) for any constant dimension and Ɛ. As a byproduct, we get the first known non-trivial upper bound on the number of Steiner points in an optimal solution to an instance of Euclidean minimum-cost Steiner biconnectivity.