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

• Volume: 36, Issue: 1, page 53-71
• ISSN: 0399-0559

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## Abstract

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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 $\left(2+ϵ\right)$-approximation algorithm for the covering problem. Finally, we show that, unless $𝒫=\mathrm{𝒩𝒫}$, the covering problem cannot be approximated in polynomial time within arbitrarily good precision.

## How to cite

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Kovaleva, Sofia, and Spieksma, Frits C. R.. "Primal-dual approximation algorithms for a packing-covering pair of problems." RAIRO - Operations Research - Recherche Opérationnelle 36.1 (2002): 53-71. <http://eudml.org/doc/245768>.

@article{Kovaleva2002,
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 $(2+\epsilon )$-approximation algorithm for the covering problem. Finally, we show that, unless $\mathcal \{P\}=\mathcal \{NP\}$, the covering problem cannot be approximated in polynomial time within arbitrarily good precision.},
author = {Kovaleva, Sofia, Spieksma, Frits C. R.},
journal = {RAIRO - Operations Research - Recherche Opérationnelle},
keywords = {primal-dual; approximation algorithms; packing-covering; intervals},
language = {eng},
number = {1},
pages = {53-71},
publisher = {EDP-Sciences},
title = {Primal-dual approximation algorithms for a packing-covering pair of problems},
url = {http://eudml.org/doc/245768},
volume = {36},
year = {2002},
}

TY - JOUR
AU - Kovaleva, Sofia
AU - Spieksma, Frits C. R.
TI - Primal-dual approximation algorithms for a packing-covering pair of problems
JO - RAIRO - Operations Research - Recherche Opérationnelle
PY - 2002
PB - EDP-Sciences
VL - 36
IS - 1
SP - 53
EP - 71
AB - 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 $(2+\epsilon )$-approximation algorithm for the covering problem. Finally, we show that, unless $\mathcal {P}=\mathcal {NP}$, the covering problem cannot be approximated in polynomial time within arbitrarily good precision.
LA - eng
KW - primal-dual; approximation algorithms; packing-covering; intervals
UR - http://eudml.org/doc/245768
ER -

## References

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1. [1] 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).
2. [2] 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. Zbl0977.68539
3. [3] 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). Zbl0937.68002MR1734026
4. [4] A. Bar–Noy, S. Guha, J. Naor and B. Schieber, Approximating the Throughput of Multiple Machines in Real-Time Scheduling. SIAM J. Comput. 31 (2001) 331-352. Zbl0994.68073
5. [5] A. Bar–Noy, R. Bar–Yehuda, A. Freund, J. Naor and B. Schieber, A Unified Approach to Approximating Resource Allocation and Scheduling. J. ACM 48 (2001) 1069-1090. Zbl1323.68564
6. [6] P. Berman and B. DasGupta, Multi-phase Algorithms for Throughput Maximization for Real-Time Scheduling. J. Combin. Optim. 4 (2000) 307-323. Zbl0991.90061MR1776667
7. [7] 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). Zbl1044.68753
8. [8] N. Garg, V.V. Vazirani and M. Yannakakis, Primal-Dual Approximation Algorithms for Integral Flow and Multicut in Trees. Algorithmica 18 (1997) 3-20. Zbl0873.68075MR1432026
9. [9] 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).
10. [10] M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs. Academic Press, San Diego, California (1980). Zbl0541.05054MR562306
11. [11] D.S. Hochbaum, Approximation algorithms for NP-hard problems. PWC Publishing Company, Boston (1997). Zbl05899342
12. [12] 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).
14. [14] F.C.R. Spieksma, On the approximability of an interval scheduling problem. J. Schedul. 2 (1999) 215-227. Zbl0938.90034MR1991780
15. [15] D.P. Williamson, Course notes Primal-Dual methods, available at http://www.research.ibm.com/people/w/williamson/#Notes

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