Hostname: page-component-848d4c4894-8bljj Total loading time: 0 Render date: 2024-06-29T00:08:21.177Z Has data issue: false hasContentIssue false
  • English
  • Français

Primal-dual approximation algorithms for a packing-covering pair of problems

Published online by Cambridge University Press:  15 July 2002

Sofia Kovaleva  and
Frits C.R. Spieksma
Sofia Kovaleva
Affiliation:
Department of Mathematics, Maastricht University, P.O. Box 616, 6200 MD Maastricht, The Netherlands; s.kovaleva@math.unimaas.nl.
Frits C.R. Spieksma
Affiliation:
Department of Applied Economics, Katholieke Universiteit Leuven, Naamsestraat 69, 3000 Leuven, Belgium.
Get access

Abstract

We consider a special packing-covering pair of problems. The packing problem is a natural generalization of finding a (weighted) maximum independent set in an interval graph, the covering problem generalizes the problem of finding a (weighted) minimum clique cover in an interval graph. The problem pair involves weights and capacities; we consider the case of unit weights and the case of unit capacities. In each case we describe a simple algorithm that outputs a solution to the packing problem and to the covering problem that are within a factor of 2 of each other. Each of these results implies an approximative min-max result. For the general case of arbitrary weights and capacities we describe an LP-based (2 + ε)-approximation algorithm for the covering problem. Finally, we show that, unless P = NP, the covering problem cannot be approximated in polynomial time within arbitrarily good precision.

Keywords

Primal-dualapproximation algorithmspacking-coveringintervals.
Type
Research Article
Information
RAIRO - Operations Research , Volume 36 , Issue 1 , January 2002 , pp. 53 - 71
Copyright
© EDP Sciences, 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

J. Aerts and E.J. Marinissen, Scan chain design for test time reduction in core-based ICs, in Proc. of the International Test Conference. Washington DC (1998).
S. Arora, C. Lund, R. Motwani, M. Sudan and M. Szegedy, Proof verification and hardness of approximation problems, in Proc. of the 33rd IEEE Symposium on the Foundations of Computer Science (1992) 14-23.
G. Ausiello, P. Crescenzi, G. Gambosi, V. Kann, A. Marchetti Spaccamela and M. Protasi, Complexity and Approximation. Combinatorial Optimization Problems and their Approximability Properties. Springer Verlag, Berlin (1999).
Bar-Noy, A., Guha, S., Naor, J. and Schieber, B., Approximating the Throughput of Multiple Machines in Real-Time Scheduling. SIAM J. Comput. 31 (2001) 331-352. CrossRef
Bar-Noy, A., Bar-Yehuda, R., Freund, A., Naor, J. and Schieber, B., Unified Approach, A to Approximating Resource Allocation and Scheduling. J. ACM 48 (2001) 1069-1090. CrossRef
Berman, P. and DasGupta, B., Multi-phase Algorithms for Throughput Maximization for Real-Time Scheduling. J. Combin. Optim. 4 (2000) 307-323. CrossRef
T. Erlebach and F.C.R. Spieksma, Simple algorithms for a weighted interval selection problem, in Proc. of the 11th Annual International Symposium on Algorithms and Computation (ISAAC '00). Lecture Notes in Comput. Sci. 1969 (2000) 228-240 (see also Report M00-01, Maastricht University).
Garg, N., Vazirani, V.V. and Yannakakis, M., Primal-Dual Approximation Algorithms for Integral Flow and Multicut in Trees. Algorithmica 18 (1997) 3-20. CrossRef
M.X. Goemans and D.P. Williamson, The primal-dual method for approximation algorithms and its application to network design problems, Chap. 4 of Approximation algorithms for NP-hard problems, edited by D.S. Hochbaum. PWC Publishing Company, Boston (1997).
M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs. Academic Press, San Diego, California (1980).
D.S. Hochbaum, Approximation algorithms for NP-hard problems. PWC PublishingCompany, Boston (1997).
D.S. Hochbaum, Approximating Covering and Packing Problems: Set Cover, Vertex Cover, Independent Set and Related Problems, Chap. 3 of Approximation algorithms for NP-hard problems, edited by D.S. Hochbaum. PWC Publishing Company, Boston (1997).
C.H. Papadimitriou, Computational Complexity. Addison-Wesley, Reading, Massachussets (1994).
Spieksma, F.C.R., On the approximability of an interval scheduling problem. J. Schedul. 2 (1999) 215-227. 3.0.CO;2-Y>CrossRef
D.P. Williamson, Course notes Primal-Dual methods, available at http://www.research.ibm.com/people/w/williamson/#Notes