# Primal-dual approximation algorithms for feedback problems in planar graphs

## Abstract

Given a subset of cycles of a graph, we consider the problem of finding 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 graphs, the subset feedback vertex set problem, and the graph bipartization 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 subset of cycles obeys certain properties. This results in 9/4-approximation algorithms for the aforementioned feedback and bipartization problems in planar graphs. Our algorithms use the primaldual method for approximation algorithms as given in Goemans and Williamson [14]. We also show that our results have an interesting bearing on a conjecture of Akiyama and Watanabe [2] on the cardinality of feedback vertex sets in planar graphs.

## Keywords

Approximation Algorithm Planar Graph Undirected Graph Network Design Problem Linear Programming Relaxation## Preview

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## References

- 1.A. Agrawal, P. Klein, and R. Ravi. When trees collide: An approximation algorithm for the generalized Steiner problem on networks.
*SIAM Journal on Computing*, 24:440–456, 1995.CrossRefGoogle Scholar - 2.Akiyama and Watanabe. Research problem.
*Graphs and Combinatorics*, 3:201–202, 1986.CrossRefGoogle Scholar - 3.M. Albertson and D. Berman. A conjecture on planar graphs. In J. Bondy and U. Murty, editors,
*Graph Theory and Related Topics*. Academic Press, 1979.Google Scholar - 4.V. Bafna, P. Berman, and T. Fujito. Constant ratio approximation of the weighted feedback vertex set problem for undirected graphs. In J. Staples, P. Eades, N. Katoh, and A. Moffat, editors,
*ISAAC '95 Algorithms and Computation*, volume 1004 of*Lecture Notes in Computer Science*, pages 142–151, 1995.Google Scholar - 5.R. Bar-Yehuda, D. Geiger, J. Naor, and R. M. Roth. Approximation algorithms for the vertex feedback set problem with applications to constraint satisfaction and Bayesian inference. In
*Proceedings of the 5th Annual ACM-SIAM Symposium on Discrete Algorithms*, pages 344–354, 1994.Google Scholar - 6.A. Becker and D. Geiger. Approximation algorithms for the loop cutset problem. In
*Proceedings of the 10th Conference on Uncertainty in Artificial Intelligence*, pages 60–68, 1994.Google Scholar - 7.O. Borodin. On acyclic colorings of planar graphs.
*Discrete Mathematics*, 25:211–236, 1979.CrossRefGoogle Scholar - 8.G. Even, J. Naor, B. Schieber, and M. Sudan. Approximating minimum feedback sets and multi-cuts in directed graphs. In E. Balas and J. Clausen, editors,
*Integer Programming and Combinatorial Optimization*, volume 920 of*Lecture Notes in Computer Science*, pages 14–28. Springer-Verlag, 1995.Google Scholar - 9.G. Even, J. Naor, B. Schieber, and L. Zosin. Approximating minimum subset feedback sets in undirected graphs with applications to multicuts. Manuscript, 1995.Google Scholar
- 10.M. R. Garey and D. S. Johnson.
*Computers and Intractability*. W.H. Freeman and Company, New York, 1979.Google Scholar - 11.N. Garg, V. Vazirani, and M. Yannakakis. Primal-dual approximation algorithms for integral flow and multicut in trees, with applications to matching and set cover. In
*Proceedings of the 20th International Colloquium on Automata, Languages and Programming*, 1993. To appear in*Algorithmica*under the title “Primal-dual approximation algorithms for integral flow and multicut in trees”.Google Scholar - 12.N. Garg, V. V. Vazirani, and M. Yannakakis. Approximate max-flow min(multi) cut theorems and their applications. In
*Proceedings of the 25th Annual ACM Symposium on Theory of Computing*, pages 698–707, 1993. To appear in*SIAM J. Comp.*Google Scholar - 13.M. X. Goemans and D. P. Wiffiamson. A general approximation technique for constrained forest problems.
*SIAM Journal on Computing*, 24: 296–317, 1995.CrossRefGoogle Scholar - 14.M. X. Goemans and D. P. Wiffiamson. The primal-dual method for approximation algorithms and its application to network design problems. In D. S. Hochbaum, editor,
*Approximation Algorithms for NP-hard Problems*, chapter 4. PWS, Boston, 1996. Forthcoming.Google Scholar - 15.D. S. Hochbaum. Good, better, best, and better than best approximation algorithms. In D. S. Hochbaum, editor,
*Approximation Algorithms for NP-hard Problems*, chapter 9. PWS, Boston, 1996. Forthcoming.Google Scholar - 16.T. R. Jensen and B. Toft.
*Graph Coloring Problems*. John Wiley and Sons, New York, 1995.Google Scholar - 17.P. Klein, S. Rao, A. Agrawal, and R. Ravi. An approximate max-flow min-cut relation for undirected multicommodity flow, with applications.
*Combinatorica*, 15:187–202, 1995.CrossRefGoogle Scholar - 18.P. Klein and R. Ravi. When cycles collapse: A general approximation technique for constrained two-connectivity problems. In
*Proceedings of the Third MPS Conference on Integer Programming and Combinatorial Optimization*, pages 39–55, 1993. Also appears as Brown University Technical Report CS-92-30.Google Scholar - 19.T. Leighton and S. Rao. An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms. In
*Proceedings of the 29th Annual Symposium on Foundations of Computer Science*, pages 422–431, 1988.Google Scholar - 20.G. L. Nemhauser and L. E. Trotter Jr. Vertex packing: Structural properties and algorithms.
*Mathematical Programming*, 8:232–248, 1975.CrossRefGoogle Scholar - 21.P. D. Seymour. Packing directed circuits fractionally.
*Combinatorica*, 15:281–288, 1995.CrossRefGoogle Scholar - 22.H. Stamm. On feedback problems in planar digraphs. In R. Möhring, editor,
*Graph-Theoretic Concepts in Computer Science*, number 484 in Lecture Notes in Computer Science, pages 79–89. Springer-Verlag, 1990.Google Scholar - 23.D. P. Williamson, M. X. Goemans, M. Mihail, and V. V. Vazirani. A primal-dual approximation algorithm for generalized Steiner network problems.
*Combinatorica*, 15:435–454, 1995.CrossRefGoogle Scholar - 24.M. Yannakakis. Node and edge-deletion NP-complete problems. In
*Proceedings of the 10th Annual ACM Symposium on Theory of Computing*, pages 253–264, May 1978.Google Scholar