An algorithm for covering polygons with rectangles
Information and Control
Augmenting graphs to meet edge-connectivity requirements
SIAM Journal on Discrete Mathematics
Network flows: theory, algorithms, and applications
Network flows: theory, algorithms, and applications
Efficient splitting off algorithms for graphs
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
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
Finding maximum flows in undirected graphs seems easier than bipartite matching
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Finding minimum generators of path systems
Journal of Combinatorial Theory Series B
Augmenting undirected edge connectivity in Õ(n2) time
Journal of Algorithms
Dilworth's Theorem and Its Application for Path Systems of a Cycle - Implementation and Analysis
ESA '99 Proceedings of the 7th Annual European Symposium on Algorithms
Incresing the Vertex-Connectivity in Directed Graphs
ESA '93 Proceedings of the First Annual European Symposium on Algorithms
Pushdown-reduce: an algorithm for connectivity augmentation and poset covering problems
Discrete Applied Mathematics
Independence free graphs and vertex connectivity augmentation
Journal of Combinatorial Theory Series B
An algorithm for (n-3)-connectivity augmentation problem: Jump system approach
Journal of Combinatorial Theory Series B
An algorithm to increase the node-connectivity of a digraph by one
Discrete Optimization
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In their seminal paper, Frank and Jordán [1995] show that a large class of optimization problems, including certain directed graph augmentation, fall into the class of covering supermodular functions over pairs of sets. They also give an algorithm for such problems, however, it relies on the ellipsoid method. Prior to our result, combinatorial algorithms existed only for the 0--1 valued problem. Our key result is a combinatorial algorithm for the general problem that includes directed vertex or S−T connectivity augmentation. The algorithm is based on Benczúr's previous algorithm for the 0--1 valued case [Benczúr 2003]. Our algorithm uses a primal-dual scheme for finding covers of partially ordered sets that satisfy natural abstract properties as in Frank and Jordán. For an initial (possibly greedy) cover, the algorithm searches for witnesses for the necessity of each element in the cover. If no two (weighted) witnesses have a common cover, the solution is optimal. As long as this is not the case, the witnesses are gradually exchanged for smaller ones. Each witness change defines an appropriate change in the solution; these changes are finally unwound in a shortest-path manner to obtain a solution of size one less.