A subset F ⊆ A [F ⊆ V] of the arcs [vertices] of a digraph D=(V,A) is called a feedback arc set (fas) [feedback vertex set (fvs)], iff D-F is an acyclic digraph (DAG). The main results in this paper are (n=|V|): (1) An O(n* log(n)) approximation algorithm is developed for the minimum-fas-problem on planar digraphs with a worst-case-ratio of 2. In the case of a planar digraph with all embeddings in the plane having at most one clockwise/anticlockwise cyclic face the given algorithm computes the optimal solution of the minimum-fas-problem. Both results hold for the weighted fas-problem, too. It will be shown that the minimum-fas-problem restricted to destroy all clockwise/anticlockwise directed simple cycles only for some fixed planar embedding of a given digraph is exactly solvable in time O(n* log(n)). (2) An O(n2) approximation algorithm is developed for the minimum-fvs-problem on planar digraphs with a worst-case-ratio bounded by min{dmax (D), CF (D)-1,2*wcrM}, where CF(D) is the number of cyclic faces in the planar embedding of D chosen by the algorithm and wcrM is the worst-case-ratio of any approximation-algorithm for the directed Steiner-Tree-Problem. So the minimum-fvs-problem for the class DPS2 of planar strongly connected digraphs having at most 2 cyclic faces in all planar embeddings becomes polynomial solvable.
CITATION STYLE
Stamm, H. (1991). On feedback problems in planar digraphs. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 484 LNCS, pp. 79–89). Springer Verlag. https://doi.org/10.1007/3-540-53832-1_33
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