An algorithm for covering polygons with rectangles
Information and Control
Augmenting graphs to meet edge-connectivity requirements
SIAM Journal on Discrete Mathematics
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
Journal of Experimental Algorithmics (JEA)
The primal-dual method for approximation algorithms and its application to network design problems
Approximation algorithms for NP-hard problems
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
Pushdown-reduce: an algorithm for connectivity augmentation and poset covering problems
Discrete Applied Mathematics
A Weighted kt, t-Free t-Factor Algorithm for Bipartite Graphs
Mathematics of Operations Research
A weighted Kt, t-free t-factor algorithm for bipartite graphs
IPCO'08 Proceedings of the 13th international conference on Integer programming and combinatorial optimization
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In their seminal paper, Frank and Jordán show that a large class of optimization problems including certain directed edge augmentation ones fall into the class of covering supermodular functions over pairs of sets. They also give an algorithm for such problems, however that 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 the second author's previous algorithm for the 0-1 valued case.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 by 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.