Abstract
A Greedy Randomized Adaptive Search Procedure (GRASP) is a randomized heuristic that has produced high quality solutions for a wide range of combinatorial optimization problems. The NP-complete Feedback Vertex Set (FVS) Problem is to find the minimum number of vertices that need to be removed from a directed graph so that the resulting graph has no directed cycle. The FVS problem has found applications in many fields, including VLSI design, program verification, and statistical inference. In this paper, we develop a GRASP for the FVS problem. We describe GRASP construction mechanisms and local search, as well as some efficient problem reduction techniques. We report computational experience on a set of test problems using three variants of GRASP.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. Bafna, P. Berman, and T. Fujito, “Approximating feedback vertex set for undirected graphs within ratio 2,” 1994, manuscript.
A. Becker and G. Geiger, “Approximation algorithms for the loop cutset problem,” in Proc. of the 10th Conference on Uncertainty in Artificial Intelligence, 1979, pp. 60–68.
P. Erdös and L. Pósa, “On the maximal number of disjoint circuits of a graph,” Publ. Math. Debrechen, vol. 9, pp. 3–12, 1962.
T.A. Feo and M.G.C. Resende, “Greedy randomized apaptive search procedures,” Journal of Global Optimization, vol. 6, pp. 109–133, 1995.
T.A. Feo, M.G.C. Resende, and S.H. Smith, “A greedy randomized adaptive search procedure for maximum independent set,” Operations Research, vol. 42, pp. 860–878, 1994.
M. Funke, 1996, personal communication.
M. Funke and G. Reinelt, “A polyhedral approach to the feedback vertex set problem,” 1996, manuscript.
M.R. Garey and D.S. Johnson, Computers and Reducibility-A Guide to the Theory of NP-Completeness, W.H. Freeman: San Francisco, 1979.
H. Levy and L. Lowe, “A contraction algorithm for finding small cycle cutsets,” Journal of Algorithms, vol. 9, pp. 470–493, 1988.
Y. Li, P.M. Pardalos, and M.G.C. Resende, “A greedy randomized adaptive search procedure for the quadratic assignment problem,” in Quadratic Assignment and Related Problems, P. Pardalos and H. Wolkowicz (Eds.), DIMACS Series on Discrete Mathematics and Theoretical Computer Science, vol. 16, American Mathematical Society, Providence, RI, 1994, pp. 237–261.
B. Monien and R. Schultz, “Four approximation algorithms for the feedback vertex set problem,” in Proc. of the 7th Conference on Graph Theoretic Concepts of Computer Science, 1981, pp. 315–390.
P.M. Pardalos, L.S. Pitsoulis, and M.G.C. Resende, “A parallel GRASP implementation for the quadratic assignment problem,” in Parallel Algorithms for Irregularly Structured Problems-Irregular'94, A. Ferreira and J. Rolim, (Eds.), Kluwer Academic Publishers, 1995, pp. 111–130.
T. Qian, Y. Ye, and P.M. Pardalos, “A pseudo "-approximation algorithm for FVS,” in State of the Art in Global Optimization, C. Floudas and P. Pardalos (Eds.), Kluwer Academic Publishers: Dordrecht, Boston, London, 1996, pp. 341–351.
M.G.C. Resende and T.A. Feo, “A GRASP for satisfiability,” in The Second DIMACS Implementation Challenge, M. Trick and D. Johnson (Eds.), DIMACS Series on Discrete Mathematics and Theoretical Computer Science, vol. 26, American Mathematical Society, 1996, pp. 499–520.
L. Schrage, “A more portable Fortran random number generator,” ACM Transactions on Mathematical Software, vol. 5, pp. 132–138, 1979.
P.D. Seymour, “Packing directed circuits fractionally,” Combinatorica, vol. 15, pp. 182–188, 1995.
A. Shamir, “Alinear time algorithm for finding minimum cutsets in reduced graphs,” SIAM Journal On Computing, vol. 8, pp. 654–655, 1979.
C. Wang, E. Lloyd, and M. Soffa, “Feedback vertex sets and cyclically reducible graphs,” Journal of the ACM, vol. 32, pp. 296–313, 1985.
M. Yannakakis, “Node and edge-deletion NP-complete problems,” in Proc. of the 10th Annual ACM Symp. on Theory of Computing, 1978, pp. 253–264.
B. 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 Proc. of the 5th Annual ACM-SIAM Symp. on Discrete Algorithms, 1994, pp. 344–354.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Pardalos, P.M., Qian, T. & Resende, M.G. A Greedy Randomized Adaptive Search Procedure for the Feedback Vertex Set Problem. Journal of Combinatorial Optimization 2, 399–412 (1998). https://doi.org/10.1023/A:1009736921890
Issue Date:
DOI: https://doi.org/10.1023/A:1009736921890