On feedback problems in planar digraphs

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{ d_max (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.