Abstract
We present a simple and unified approach for developing and analyzing approximation algorithms for covering problems. We illustrate this on approximation algorithms for the following problems: Vertex Cover, Set Cover, Feedback Vertex Set, Generalized Steiner Forest and related problems.
The main idea can be phrased as follows: iteratively, pay two dollars (at most) to reduce the total optimum by one dollar (at least), so the rate of payment is no more than twice the rate of the optimum reduction. This implies a total payment (i.e., approximation cost) ≤ twice the optimum cost.
Our main contribution is based on a formal definition for covering problems, which includes all the above fundamental problems and others. We further extend the Bafna, Berman and Fujito Local-Ratio theorem. This extension eventually yields a short generic r-approximation algorithm which can generate most known approximation algorithms for most covering problems.
Another extension of the Local-Ratio theorem to randomized algorithms gives a simple proof of Pitt's randomized approximation for Vertex Cover. Using this approach, we develop a modified greedy algorithm, which for Vertex Cover, gives an expected performance ratio ≤ 2.
This research was supported by the fund for the promotion of research at the Technion.
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A. Agrawal, P. Klein, and R. Ravi. When trees collide: an approximation algorithm for the generalized steiner problem in networks. Proc. 23rd ACM Symp. on Theory of Computing, pages 134–144, 1991.
V. Bafna, P. Berman, and T. Fujito. Constant ratio approximation of the weighted feedback vertex set problem for undirected graphs. ISAAC '95 Algorithms and Computation, (1004):142–151, 1995.
R. Bar-Yehuda and S. Even. A linear time approximation algorithm for the weighted vertex cover problem. Journal of Algorithms, 2:198–203, 1981.
R. Bar-Yehuda and S. Even. A local-ratio theorem for approximating the weighted vertex cover problem. Annals of Discrete Mathematics, 25:27–46, 1985.
R. Bar-Yehuda. A linear time 2-approximation algorithm for the min clique-complement problem. Technical Report CS0933, Technion Haifa, May 1998.
R. Bar-Yehuda. Partial vertex cover problem and its generalizations. Technical Report CS0934, Technion Haifa, May 1998.
R. Bar-Yehuda, D. Geiger, J. Naor, and R. Roth. Approximation algorithms for the vertex feedback set problem with applications to constraint satisfaction and bayesian inference. Accepted to SIAM J. on Computing, 1997.
R. Bar-Yehuda and D. Rawitz. Generalized algorithms for bounded integer pro-grams with two variables per constraint. Technical Report CS0935, Technion Haifa, May 1998.
A. Becker and D. Geiger. Approximation algorithms for the loop cutset problem. Proc. 10th Conf. on Uncertainty in Artificial Intelligence, pages 60–68, 1994.
F. Chudak, M. Goemans, D. Hochbaum, and D. Williamson. A primal-dual interpretation of recent 2-approximation algorithms for the feedback vertex set problem in undirected graphs. Unpublished, 1996.
V. Chvatal. A greedy heuristic for the set-covering problem. Math. of Oper. Res., 4(3):233–235, 1979.
K. Clarkson. A modification of the greedy algorithm for the vertex cover. Info. Proc. Lett., 16:23–25, 1983.
T. Fujito. A unified local ratio approximation of node-deletion problems. ESA, Barcelona, Spain, pages 167–178, September 1996.
M. Garey and D. Johnson. Computers and Intractability. W.H. Freeman, 1979.
M. Goemans and D. Williamson. A general approximation technique for constrained forest problems. SIAM Journal on Computing, 24(2):296–317, 1995.
M. Goemans and D. Williamson. The primal-dual method for approximation algorithms and its application to network design problems. Approximation Algorithms for NP-Hard Problems, 4, 1996.
D. Hochbaum. Approximating covering and packing problems: Set cover, vertex cover, independent set, and related problems. Chapter 3 in Approximation Algorithms for NP-Hard Problems, PWS Publication Company, 1997.
G. Nemhauser and J. L.E. Trotter. Vertex packings: structural properties and algorithms. Mathematical Programming, 8:232–248, 1975.
V. T. Paschos. A survey of approximately optimal solutions to some covering and packing problems. ACM Computing Surveys, 29(2):171–209, June 1997.
L. Pitt. Simple probabilistic approximation algorithm for the vertex cover problem. Technical Report, Yale, June 1984.
D. Williamson, M. Goemans, M. Mihail, and V. Vazirani. A primal-dual approximation algorithm for generalized steiner network problems. Combinatorica, 15:435–454, 1995.
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© 1998 Springer-Verlag Berlin Heidelberg
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Bar-Yehuda, R. (1998). One for the price of two: A unified approach for approximating covering problems. In: Jansen, K., Rolim, J. (eds) Approximation Algorithms for Combinatiorial Optimization. APPROX 1998. Lecture Notes in Computer Science, vol 1444. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0053963
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DOI: https://doi.org/10.1007/BFb0053963
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