PRIMAL-DUAL APPROXIMATION ALGORITHMS FOR FEEDBACK PROBLEMS IN PLANAR GRAPHS

Given a subset of cycles of a graph, we consider the problem of findin g a minimum-weight set of vertices that meets all cycles in the subset . This problem generalizes a number of problems, including the minimum -weight feedback vertex set problem in both directed and undirected gr aphs, the subset feedback vertex set problem, and the graph bipartizat ion problem, in which one must remove a minimum-weight set of vertices so that the remaining graph is bipartite. We give a 9/4-approximation algorithm for the general problem in planar graphs, given that the su bset of cycles obeys certain properties. This results in 9/4-approxima tion algorithms for the aforementioned feedback and bipartization prob lems in planar graphs. Our algorithms use the primal-dual method for a pproximation algorithms as given in Goemans and Williamson [16]. We al so show that our results have an interesting bearing on a conjecture o f Akiyama and Watanabe [2] on the cardinality of feedback vertex sets in planar graphs.