Abstract
A subset F\(\subseteq\)A [F\(\subseteq\)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.
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© 1991 Springer-Verlag Berlin Heidelberg
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Stamm, H. (1991). On feedback problems in planar digraphs. In: Möhring, R.H. (eds) Graph-Theoretic Concepts in Computer Science. WG 1990. Lecture Notes in Computer Science, vol 484. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-53832-1_33
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DOI: https://doi.org/10.1007/3-540-53832-1_33
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