Abstract
We overview recent results on the existence of polynomial time approximation schemes for some dense instances of NP-hard optimization problems. We indicate further some inherent limits for existence of such schemes for some other dense instances of the optimization problems.
Research partially supported by the International Computer Science Institute, Berkeley, California, by the DFG Grant KA 673 4-1, and by the ESPRIT BR Grants 7097 and EC-US 030, by DIMACS, and by the Max-Planck Research Prize.
Preview
Unable to display preview. Download preview PDF.
References
N. Alon, A. Frieze and D. Welsh, Polynomial Time Randomized Approximation Schemes for the Tutte Polynomial of Dense Graphs, Random Structures and Algorithms 6 (1995), pp. 459–478.
S. Arora, A. Frieze and H. Kaplan, A New Rounding Procedure for the Assignment Problem with Applications to Dense Graph Arrangements, Proc. 37th IEEE FOCS (1996), pp. 21–30.
S. Arora, D. Karger, and M. Karpinski, Polynomial Time Approximation Schemes for Dense Instances of NP-Hard Problems, Proc. 27th ACM STOC (1995), pp. 284–293.
S. Arora and C. Lund, Hardness of Approximations, in Approximation Algorithms for NP-Hard Problems (D. Hochbaum, ed.), PWS Publ. Co. (1997), pp. 399–446.
S. Arora, C. Lund, R. Motwani, M. Sudan and M. Szegedy, Proof Verification and Hardness of Approximation Problems, Proc. 33rd IEEE FOCS (1992), pp. 14–20.
M. Bern and P. Plassmann, The Steiner Problem with Edge Lengths 1 and 2, Inform. Process. Lett. 32 (1989), pp. 171–176.
A.Z. Broder, How Hard is it to Marry at Random (On the Approximation of the Permanent), Proc. 18th ACM STOC (1986), pp. 50–58, Erratum in Proc. 20th ACM STOC (1988), p. 551.
P. Chinn, J. Chvatalova, A. Dewdney, N. Gibbs, The Bandwidth Problem for Graphs and Matrices — A Survey, Journal of Graph Theory 6 (1982), pp. 223–254.
A. Clementi and L. Trevisan, Improved Nonapproximability Result for Vertex Cover with Density Constraints, Proc. 2nd Int. Conference, COCOON '96, Springer-Verlag (1996), pp. 333–342.
J. Diaz, M. Serna and P. Spirakis, Some Remarks on the Approximability of Graph Layout Problems, Technical Report LSI-94-16-R, Univ. Politec, Catalunya (1994).
M.E. Dyer, A. Frieze and M.R. Jerrum, Approximately Counting Hamilton Cycles in Dense Graphs, Proc. 4th ACM-SIAM SODA (1994), pp. 336–343.
U. Feige, A Threshold of In n for Approximating Set Cover, Proc. 28th ACM STOC (1996), pp. 314–318.
W. Fernandez-de-la-Vega, MAX-CUT has a Randomized Approximation Scheme in Dense Graphs, Random Structures and Algorithms 8 (1996), pp. 187–999.
W. Fernandez-de-la-Vega and M. Karpinski, Polynomial Time Approximability of Dense Weighted Instances of MAX-CUT, Research Report No. 85171-CS, University of Bonn (1997).
W. Fernandez-de-la-Vega, G. S. Lueker, Bin Packing Can be Solved Within 1+∈ in Linear Time, Combinatorica 1 (1981), pp. 349–355.
A. Frieze and R. Kannan, The Regularity Lemma and Approximation Schemes for Dense Problems, Proc. 37th IEEE FOCS (1996), pp. 12–20.
A. Frieze and R. Kannan, Quick Approximation to Matrices and Applications, Manuscript (1997).
M. Garey, R. Graham, D. Johnson, D. Knuth, Complexity Results For Bandwidth Minimization, SIAM J. Appl. Math. 34 (1978), pp. 477–495.
M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman (1979).
M. Goemans and D. Williamson, 878-approximation Algorithms for MAX-CUT and MAX2SAT, Proc. 26th ACM STOC (1994), pp. 422–431.
O. Goldreich, S. Goldwasser and D. Ron, Property Testing and its Connection to Learning and Approximation, Proc. 37th IEEE FOCS (1996), pp. 339–348.
J. Haralambides, F. Makedon, B. Monien, Bandwidth Minimization: An Approximation Algorithm for Caterpillars, Math. Systems Theory 24 (1991), pp. 169–177.
J. Håstad, Clique is Hard to Approximate within n 1−∈, Proc. 37th IEEE FOCS (1986), pp. 627–636.
D. Hochbaum, Approximating Covering and Packing Problems: Set Cover, Vertex Cover, Independent Set, and Related Problems, in Approximation Algorithms for NP-hard Problems (D. Hochbaum, ed.), PWS Publ. Co. (1997), pp. 94–143.
O. H. Ibarra, C. E. Kim, Fast Approximation Algorithms for the Knapsack and Sum of Subsets Problems, J. ACM 22 (1975), pp. 463–468.
M. R. Jerrum and A. Sinclair, Approximating the Permanent, SIAM J. Comput. 18 (1989), pp. 1149–1178.
D. S. Johnson, Approximation Algorithms for Combinatorial Problems, J. Comput. System Sciences 9 (1974), pp. 256–278.
N. Karmarkar and R. M. Karp, An Efficient Approximation Scheme for the One-dimensional Bin-Packing Problem, Proc. 23rd IEEE FOCS (1982), pp. 312–320.
R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (R. Miller and J. Thatcher, ed.), Plenum Press (1972), pp. 85–103.
M. Karpinski, J. Wirtgen and A. Zelikovsky, An Approximation Algorithm for the Bandwidth Problem on Dense Graphs, ECCC Technical Report TR 97-017 (1997).
M. Karpinski and A. Zelikovsky, Approximating Dense Cases of Covering Problems (Preliminary Version), Technical Report TR-96-059, International Computer Science Institute, Berkeley (1996).
M. Karpinski and A. Zelikovsky, New Approximation Algorithms for the Steiner Tree Problem, J. of Combinatorial Optimization 1 (1997), pp. 47–65.
M. Karpinski and A. Zelikovsky, Approximating Dense Cases of Covering Problems, ECCC Technical Report TR 97-004, 1997, to appear in Proc. DIMACS Workshop on Network Design: Connectivity and Facilities Location, Princeton (1997).
G. Kortsarz and D. Peleg, On Choosing a Dense Subgraph, Proc. 34th IEEE FOCS (1993), pp. 692–701.
B. Monien, The Bandwidth Minimization Problem for Caterpillars with Hair Length 3 is NP-Complete. SIAM J. Alg. Disc. Math. 7 (1986), pp. 505–514.
C. Papadimitriou, Computational Complexity, Addison-Wesley, (1994).
C. Papadimitriou and M. Yannakakis, Optmization, Approximation and Complexity Classes, J. Comput. System Sciences 43 (1991), pp. 425–440.
C. Papadimitriou and M. Yannakakis, On Limited Nondeterminism and the Complexity of the VC-dimension, J. Comput. System Sciences 53 (1996), pp. 161–170.
P. Raghavan, Probabilistic Construction of Deterministic Algorithms: Approximate Packing Integer Programs, J. Comput. System Sciences 37 (1988), pp. 130–143.
L. Smithline, Bandwidth of the Complete k-ary Tree, Discrete Mathematics 142 (1995), pp. 203–212.
M. Yannakakis, On the Approximation of Maximum Satisfiability, Proc. 3rd ACM-SIAM SODA (1992), pp. 1–9.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Karpinski, M. (1997). Polynomial time approximation schemes for some dense instances of NP-hard optimization problems. In: Rolim, J. (eds) Randomization and Approximation Techniques in Computer Science. RANDOM 1997. Lecture Notes in Computer Science, vol 1269. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63248-4_1
Download citation
DOI: https://doi.org/10.1007/3-540-63248-4_1
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-63248-1
Online ISBN: 978-3-540-69247-8
eBook Packages: Springer Book Archive