Abstract
The inapproximability of non NP-hard optimization problems is investigated. Based on self-reducibility and approximation preserving reductions, it is shown that problems Log Dominating Set, Tournament Dominating Set and Rich Hypergraph Vertex Cover cannot be approximated to a constant ratio in polynomial time unless the corresponding NP-hard versions are also approximable in deterministic subexponential time. A direct connection is established between non NP-hard problems and a PCP characterization of NP. Reductions from the PCP characterization show that Log Clique is not approximable in polynomial time and Max Sparse SAT does not have a PTAS under the assumption that SAT cannot be solved in deterministic \( 2^{O(log n \sqrt n )} \)time and that NP \( \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \not\subset } \) DTIME\( (2^{O(n)} ) \).
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.
Similar content being viewed by others
References
Arora, S.: Personal Communication.
Arora, S., Lund, C.: Hardness of Approximations. In Approximation Algorithms for NP-hard problems. Ed. Dorit Hochbaum. (1997).
Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Verification and hardness of approximation problems. Proc. 33rd Ann. IEEE Symp. on Foundations of Computer Science. (1992) 14–23.
Arora, S., Safra, S.: Probabilistic Checking of Proofs. Proceedings of the 33rd IEEE Annual Symposium on Foundations of Computer Science. (1992) 2–13.
Buss, J. F., Goldsmith, J.: Nondeterminism within P. SIAM Journal on Computing. 22 (1993) 560–572.
Bellare, M., Goldwasser, S., Lund, C., Russell, A.: Efficient probabilistic checking of proofs and applications to approximation. Proceedings of the 25th Annual ACM Symposium on the Theory of Computing (1993) 113–131.
Cai, L., Chen, J.: On the amount of nondeterminism and the power of verifying. SIAM Journal on Computing. 26 (1997) 733–750
Cai, L., Chen, J.: On Fixed-Parameter Tractability and Approximability of NP Optimization Problems. Journal of Computer and System Sciences. 54 (1997) 465–474.
Cai, L., Chen, J., Downey, R., Fellows, M.: On the Structure of Parameterized Problems in NP. Information and Computation. 123 (1995) 38–49.
Díaz, J., Torán, J.: Classes of bounded nondeterminism. Math. System Theory. 23 (1990) 21–32.
Downey, R., Fellows, M.: Fixed-Parameter Intractability. Proceedings of the 7th IEEE Annual Conference on Structure in Complexity Theory. (1992) 36–49.
Downey, R., Fellows, M.: Fixed-Parameter Tractability and Completeness I: Basic Results. SIAM Journal on Computing. 24 (1995) 873–921.
Downey, R., Fellows, M.: Parameterized Computational Feasibility. The Proceedings of FEASMATH: Feasible Mathematics: A Mathematical Sciences Institute Workshop (1995).
Farr, G.: On problems with short certificates. Acta Informatica. 31 (1994) 479–502.
Feige, U.: A Threshold of ln n for Approximating Set Cover. Proceedings of The 28th Annual ACM Symposium On The Theory Of Computing. (1996) 314–318.
Feige, U., Goldwasser, S., Lovász, L., Safra, S., Szegedy M.: Approximating clique is almost NP-complete. Proceedings of the 32nd IEEE Symposium on the Foundations of Computer Science. (1991) 2–12.
Feige, U., Kilian, J.: On Limited versus Polynomial Nondeterminism. CJTCS: Chicago Journal of Theoretical Computer Science. (1997).
Fotakis, D., Spirakis, P.: (poly(loglogn), poly(loglogn))-Restricted Verifiers are Unlikely to exist for languages in NP. Proceedings of the 23nd International Symposium on Mathematical Foundations of Computer Science. (1996) 360–371.
Goldsmith J., Levy, M., Mundhenk, M.: Limited Nondeterminism. SIGACT News. 27 (1996) 20–29.
Johnson, D. S.: The NP-Completeness Column: Ongoing Guide. Journal of Algorithms. 13 (1992) 502–524.
Kann, V.: On the Approximability of NP-complete Optimization Problems. Royal Institute of Technology, Sweden. Ph.D. Thesis (1992).
Megiddo, N., Vishkin, U.: On Finding a Minimum Dominating Set in a Tournament. Theoretical Computer Science. 61 (1988) 307–316.
Papadimitriou C. H., Yannakakis M.: On Limited Nondeterminism and the Complexity of the V-C Dimension. Proceedings of the 8th Annual Conference on Structure in Complexity Theory. (1993) 12–18.
Pippenger, N., Fischer, M. J.: Relations Among Complexity Measures. Journal of the ACM. 26 (1979) 361–381.
Polishchuk, A., Spielman, D. A.: Nearly-linear Size Holographic Proofs. Proceedings of the 26th Annual ACM Symposium on the Theory of Computing. (1994) 194–203.
Szelepcsényi, R.: β k-complete problems and greediness. Computer Science Department, University, Rochester, New York. Technical Report 445 (1993).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cai, L., Juedes, D., Kanj, I. (1998). The Inapproximability of Non NP-hard Optimization Problems. In: Chwa, KY., Ibarra, O.H. (eds) Algorithms and Computation. ISAAC 1998. Lecture Notes in Computer Science, vol 1533. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-49381-6_46
Download citation
DOI: https://doi.org/10.1007/3-540-49381-6_46
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-65385-1
Online ISBN: 978-3-540-49381-5
eBook Packages: Springer Book Archive