Abstract
We present two new combinatorial approximation frameworks that are not based on LP-duality, or even on linear programming. Instead, they are based on weight manipulation in the spirit of the local ratio technique. We show that the first framework is equivalent to the LP based method of dual fitting and that the second framework is equivalent an LP-based method which we define and call primal fitting.
Our equivalence results are not unlike the recently proven equivalence between the (combinatorial) local ratio technique and the (LP based) primal-dual schema. Although equivalent, the local ratio approach is more intuitive, and indeed, several breakthrough results were first obtained within the local ratio framework and only later translated into primal-dual algorithms. We believe that our frameworks may have a similar role with respect to their LP-based counterparts (dual fitting and primal fitting). Also, we think that the notion of primal fitting is of independent interest.
We demonstrate the frameworks by presenting alternative analyses to the greedy algorithm for the set cover problem, and to known algorithms for the metric uncapacitated facility location problem. Also, we analyze a 9-approximation algorithm for a disk cover problem that arises in the design phase of cellular telephone networks.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bafna, V., Berman, P., Fujito, T.: A 2-approximation algorithm for the undirected feedback vertex set problem. SIAM Journal on Discrete Mathematics 12(3), 289–297 (1999)
Bar-Noy, A., Bar-Yehuda, R., Freund, A., Naor, J., Shieber, B.: A unified approach to approximating resource allocation and schedualing. Journal of the ACM 48(5), 1069–1090 (2001)
Bar-Noy, A., Guha, S., Katz, Y., Naor, J., Schieber, B., Shachnai, H.: Throughput maximization of real-time scheduling with batching. In: 13th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 742–751 (2002)
Bar-Yehuda, R.: One for the price of two: A unified approach for approximating covering problems. Algorithmica 27(2), 131–144 (2000)
Bar-Yehuda, R., Even, S.: A local-ratio theorem for approximating the weighted vertex cover problem. Annals of Discrete Mathematics 25, 27–46 (1985)
Bar-Yehuda, R., Rawitz, D.: On the equivalence between the primal-dual schema and the local-ratio technique. In: Goemans, M.X., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) RANDOM 2001 and APPROX 2001. LNCS, vol. 2129, pp. 24–35. Springer, Heidelberg (2001)
Becker, A., Geiger, D.: Optimization of Pearl’s method of conditioning and greedy-like approximation algorithms for the vertex feedback set problem. Artificial Intelligence 83(1), 167–188 (1996)
Bertsimas, D., Teo, C.: From valid inequalities to heuristics: A unified view of primal-dual approximation algorithms in covering problems. Operations Research 46(4), 503–514 (1998)
Chudak, F.A., Goemans, M.X., Hochbaum, D.S., Williamson, D.P.: A primaldual interpretation of recent 2-approximation algorithms for the feedback vertex set problem in undirected graphs. Operations Research Letters 22, 111–118 (1998)
Chuzhoy, J.: Private Communication
Chvátal, V.: A greedy heuristic for the set-covering problem. Mathematics of Operations Research 4(3), 233–235 (1979)
Goemans, M.X., Williamson, D.P.: The primal-dual method for approximation algorithms and its application to network design problems. In: Hochbaum [14], ch. 4
Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. Journal of Algorithms 31(1), 228–248 (1999)
Hochbaum, D.S. (ed.): Approximation Algorithms for NP-Hard Problem. PWS Publishing Company (1997)
Jain, K., Mahdian, M., Markakis, E., Saberi, A., Vazirani, V.: A greedy facility location algorithm analyzed using dual-fitting with factor-revealing LP. Journal of the ACM (2003) (to appear)
Jain, K., Vazirani, V.V.: Approximation algorithms for metric facility location and k-median problems using the primal-dual schema and Lagrangian relaxation. Journal of the ACM 48(2), 274–296 (2001)
Lovász, L.: On the ratio of optimal integral and fractional covers. Discreate Mathematics 13, 383–390 (1975)
Mahdian, M., Ye, Y., Zhang, J.: Improved approximation algorithms for metric facility location problems. In: Jansen, K., Leonardi, S., Vazirani, V.V. (eds.) APPROX 2002. LNCS, vol. 2462, pp. 229–242. Springer, Heidelberg (2002)
Shmoys, D.: Approximation algorithms for facility location problems. In: Jansen, K., Khuller, S. (eds.) APPROX 2000. LNCS, vol. 1913, pp. 27–33. Springer, Heidelberg (2000)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Freund, A., Rawitz, D. (2004). Combinatorial Interpretations of Dual Fitting and Primal Fitting. In: Solis-Oba, R., Jansen, K. (eds) Approximation and Online Algorithms. WAOA 2003. Lecture Notes in Computer Science, vol 2909. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24592-6_11
Download citation
DOI: https://doi.org/10.1007/978-3-540-24592-6_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-21079-5
Online ISBN: 978-3-540-24592-6
eBook Packages: Springer Book Archive