Abstract
A remarkable achievement of the theory of exact algorithms is that it has provided a fairly complete characterization1 of the intrinsic complexity of natural computational problems, modulo some strongly believed conjectures. Recent impressive developments raise hopes that we will some day have a comprehensive understanding of the approximability of NP-hard optimization problems as well. In this chapter we will give a brief overview of these developments.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Notes
U. Feige, S. Goldwasser, L. Lovâsz, S. Safra, and M. Szegedy. Approximating clique is almost NP-complete. In Proc. 32nd IEEE Annual Symposium on Foundations of Computer Science, pages 2–12, 1991.
L. Babai. Trading group theory for randomness. In Proc. 17th ACM Symposium on the Theory of Computing, pages 421–429, 1985.
S. Goldwasser, S. Micali, and C. Rackoff. The knowledge complexity of interactive proofs. SIAM Journal on Computing, 18: 186–208, 1989.
M. Blum and S. Kannan. Designing programs that check their work. In Proc. 21st ACM Symposium on the Theory of Computing, pages 86–97, 1989.
M. Blum, M. Luby, and R. Rubinfeld. Testing/correcting with applications to numerical problems. Journal of Computer and System Sciences, 47: 549–595, 1993.
S. Arora and S. Safra. Probabilistic checking of proofs: a new characterization of NP. In Proc. 33rd IEEE Annual Symposium on Foundations of Computer Science, pages 2–13, 1992.
S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy. Proof verification and intractability of approximation problems. In Proc. 33rd IEEE Annual Symposium on Foundations of Computer Science, pages 13–22, 1992.
J. Hastad. Some optimal inapproximability results. In Proc. 29th ACM Symposium on the Theory of Computing, pages 1–10, 1997.
C.H. Papadimitriou and M. Yannakakis. Optimization, approximation, and complexity classes. Journal of Computer and System Sciences, 43: 425–440, 1991.
M. Bern and P. Plassmann. The Steiner problem with edge lengths 1 and 2. Information Processing Letters, 32: 171–176, 1989.
A. Lubotzky, R. Phillips, and P. Sarnak. Ramanujan graphs. Combinatorica, 8: 261–277, 1988.
R. Impagliazzo and D. Zuckerman. How to recycle random bits. In Proc. 30st IEEE Annual Symposium on Foundations of Computer Science, pages 248–253, 1989.
U. Feige, S. Goldwasser, L. Lovâsz, S. Safra, and M. Szegedy. Approximating clique is almost NP-complete. In Proc. 32nd IEEE Annual Symposium on Foundations of Computer Science, pages 2–12, 1991.
J. Hastad. Clique is hard to approximate within n1-E. In Proc. 37th IEEE Annual Symposium on Foundations of Computer Science, pages 627–636, 1996.
R. Boppana and M.M. Halldôrsson. Approximating maximum independent sets by excluding subgraphs. BIT, 32: 180–196, 1992.
C. Lund and M. Yannakakis. On the hardness of approximating minimization problems. Journal of the ACM, 41: 960–981, 1994.
U. Feige. A treshold of In n for approximating set cover. Journal of the ACM, 45: 634–652, 1998.
M. Ben-or, S. Goldwasser, J. Kilian, and A. Wigderson. Multi-prover interactive proofs: How to remove intractability. In Proc. 20th ACM Symposium on the Theory of Computing, pages 113–131, 1988.
R. Raz. A parallel repetition theorem. SIAM Journal on Computing, 27: 763803, 1998.
S.K. Sahni and T.F. Gonzalez. P-complete approximation problems. Journal of the ACM, 23: 555–565, 1976.
M. Naor, L. Schulman, and A. Srinivasan. Splitters and near-optimal de-randomization. In Proc. 36th IEEE Annual Symposium on Foundations of Computer Science, pages 182–191, 1995.
H. Karloff and U. Zwick. A 7/8-approximation algorithm for MAX-3SAT? In Proc. 38th IEEE Annual Symposium on Foundations of Computer Science, pages 406–415, 1997.
S. Arora and C. Lund. Hardness of approximations. In D.S. Hochbaum, editor, Approximation Algorithms for NP-Hard Problems, pages 46–93. PWS Publishing, Boston, MA, 1997.
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.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Vazirani, V.V. (2003). Hardness of Approximation. In: Approximation Algorithms. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04565-7_29
Download citation
DOI: https://doi.org/10.1007/978-3-662-04565-7_29
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-08469-0
Online ISBN: 978-3-662-04565-7
eBook Packages: Springer Book Archive