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

Kovaleva, Sofia; Spieksma, Frits C. R.

RAIRO - Operations Research - Recherche Opérationnelle, Tome 36 (2002) no. 1, pp. 53-71.

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 $𝒫=\mathrm{𝒩𝒫}$, the covering problem cannot be approximated in polynomial time within arbitrarily good precision.

Classification : 05A05
Mots clés : primal-dual, approximation algorithms, packing-covering, intervals

[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] 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. | Zbl

[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). | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[6] P. Berman and B. Dasgupta, Multi-phase Algorithms for Throughput Maximization for Real-Time Scheduling. J. Combin. Optim. 4 (2000) 307-323. | MR | Zbl

[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). | MR | Zbl

[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. | MR | Zbl

[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] M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs. Academic Press, San Diego, California (1980). | MR | Zbl

[11] D.S. Hochbaum, Approximation algorithms for NP-hard problems. PWC Publishing Company, Boston (1997).

[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).

[13] C.H. Papadimitriou, Computational Complexity. Addison-Wesley, Reading, Massachussets (1994). | MR | Zbl

[14] F.C.R. Spieksma, On the approximability of an interval scheduling problem. J. Schedul. 2 (1999) 215-227. | MR | Zbl

[15] D.P. Williamson, Course notes Primal-Dual methods, available at http://www.research.ibm.com/people/w/williamson/#Notes