Hardness of Set Cover with Intersection 1
We consider a restricted version of the general Set Covering problem in which each set in the given set system intersects with any other set in at most 1 element. We show that the Set Covering problem with intersection 1 cannot be approximated within a o(logn) factor in random polynomial time unless NP ⊆ ZTIME(n o(log logn)). We also observe that the main challenge in derandomizing this reduction lies in finding a hitting set for large volume combinatorial rectangles satisfying certain intersection properties. These properties are not satisfied by current methods of hitting set construction.
An example of a Set Covering problem with the intersection 1 property is the problem of covering a given set of points in two or higher dimensions using straight lines; any two straight lines intersect in at most one point. The best approximation algorithm currently known for this problem has an approximation factor of θ(logn), and beating this bound seems hard. We observe that this problem is Max-SNP-Hard.
KeywordsPolynomial Time Proof System Approximation Factor Full Version Auxiliary System
Unable to display preview. Download preview PDF.
- 1.N. Alon, O. Goldreich, J. Hastad, R. Perralta. Simple Constructions of Almost k-Wise Independent Random Variables. Random Structures and Algorithms, 3, 1992.Google Scholar
- 2.V.S. Anil Kumar and H. Ramesh. Covering Rectilinear Polygons with Axis-Parallel Rectangles. Proceedings of 31st ACM-SIAM Symposium in Theory of Computing, 1999.Google Scholar
- 3.S. Arora, C. Lund. Hardness of Approximation. In Approximation Algorithms for NP-Hard Problems, Ed. D. Hochbaum, PWS Publishers, 1995, pp. 399–446.Google Scholar
- 4.S. Arora, C. Lund, R. Motwani, M. Sudan, M. Szegedy. Proof Verification and Intractability of Approximation Problems. Proceedings of 33rd IEEE Symposium on Foundations of Computer Science, 1992, pp. 13–22.Google Scholar
- 5.S. Arora, M. Sudan. Improved Low Degree Testing and Applications. Proceedings of the ACM Symposium on Theory of Computing, 1997, pp. 485–495.Google Scholar
- 7.M. Bellare, S. Goldwasser, C. Lund, A. Russell. Efficient Probabilistically Checkable Proofs and Applications to Approximation, Proceedings of 25th ACM Symposium on Theory of Computing, 1993, pp. 294–303.Google Scholar
- 13.C. Levcopoulos. Improved Bounds for Covering General Polygons by Rectangles. Proceedings of 6th Foundations of Software Tech. and Theoretical Comp. Sc. LNCS 287, 1987.Google Scholar
- 14.N. Linial, M. Luby, M. Saks, D. Zuckerman. Hitting Sets for Combinatorial Rectangles. Proceedings of 25 ACM Symposium on Theory of Computing, 1993, pp. 286–293.Google Scholar
- 15.C. Lund, M. Yannakakis. On the Hardness of Approximating Minimization Problems. Proceedings of 25th ACM Symposium on Theory of Computing, 1993, pp. 286–293.Google Scholar
- 17.M. Naor, L. Schulman, A. Srinivasan. Splitters and Near-Optimal Derandomization. Proceedings of the 36th IEEE Symposium on Foundations of Computer Science, 1995, pp. 182–191.Google Scholar
- 18.R. Raz. A Parallel Repetition Theorem. Proceedings of the 27th ACM Symposium on Theory of Computing, 1995, pp. 447–456.Google Scholar
- 19.R. Raz and S. Safra. A Sub-Constant Error-Probability Low-Degree test and a Sub-Constant Error-Probability PCP Characterization of NP. Proceedings of the ACM Symposium on Theory of Computing, 1997, pp. 475–484.Google Scholar
- 20.Madhu Sudan. Personal communication.Google Scholar