Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-23T06:53:20.388Z Has data issue: false hasContentIssue false
  • English
  • Français

An Average Case NP-complete Graph Colouring Problem

Published online by Cambridge University Press:  02 April 2018

LEONID A. LEVIN  and
RAMARATHNAM VENKATESAN
LEONID A. LEVIN
Affiliation:
Department of Computer Science, Boston University, 111 Cummington Mall, Boston, MA 02215, USA (home page: www.cs.bu.edu/fac/lnd/)
RAMARATHNAM VENKATESAN
Affiliation:
Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

NP-complete problems should be hard on some instances but those may be extremely rare. On generic instances many such problems, especially related to random graphs, have been proved to be easy. We show the intractability of random instances of a graph colouring problem: this graph problem is hard on average unless all NP problems under all samplable (i.e. generatable in polynomial time) distributions are easy. Worst case reductions use special gadgets and typically map instances into a negligible fraction of possible outputs. Ours must output nearly random graphs and avoid any super-polynomial distortion of probabilities. This poses significant technical difficulties.

Keywords

Primary 60C05Secondary 68Q1768Q2505C1505C80
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 27 , Issue 5 , September 2018 , pp. 808 - 828
Copyright
Copyright © Cambridge University Press 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†An abstract of a 20-colour version of this result is in [31] and updates in arXiv:0112001.

§

This work was supported by NSF grant CCF-1049505.

References

[1] Ajtai, M. (1996) Generating hard instances of lattice problems. In STOC '96: 28th Annual ACM Symposium on Theory of Computing, ACM, pp. 99–108.Google Scholar
[2] Aharonov, D. and Regev, O. (2005) Lattice problems in NP ∩ coNP. J. Assoc. Comput. Mach. 52 749–765.Google Scholar
[3] Angluin, D. and Valiant, L. G. (1979) Fast probabilistic algorithms for Hamilton circuits and matchings. J. Comput. Syst. Sci. 18 155–193.Google Scholar
[4] Babai, L., Erdos, P. and Selkow, M. (1980) Random graph isomorphism. SIAM J. Comput. 9 628–635.Google Scholar
[5] Begelfor, E., Miller, S. D. and Venkatesan, R. (2015) Non-Abelian analogs of lattice rounding. Groups Complexity Cryptology 7 117–133.Google Scholar
[6] Ben-David, S., Chor, B., Goldreich, O. and Luby, M. (1989) On the theory of average case complexity. In STOC '89: 21st Annual ACM Symposium on Theory of Computing, ACM, pp. 204–216.Google Scholar
[7] Blass, A. and Gurevich, Y. (1995) Matrix transformation is complete for the average case. SIAM J. Comput. 24 3–29.Google Scholar
[8] Blum, M. and Micali, S. (1984) How to generate cryptographically strong sequences of pseudo random bits. SIAM J. Comput. 13 850–864.Google Scholar
[9] Bollobás, B. (2001) Random Graphs, second edition, Cambridge University Press.Google Scholar
[10] Bollobás, B. (2004) Extremal Graph Theory, Dover.Google Scholar
[11] Goldreich, O., Goldwasser, S. and Micali, S. (1986) How to construct random functions. J. Assoc. Comput. Mach. 33 792–807.Google Scholar
[12] Gurevich, Y. (1987) Complete and incomplete randomized NP problems. In SFCS '87: 28th Annual Symposium on Foundations of Computer Science, IEEE, pp. 111–117.Google Scholar
[13] Gurevich, Y. and Shelah, S. (1987) Expected computation time for Hamilton path problem. SIAM J. Comput. 16 486–502.Google Scholar
[14] Gurevich, Y. (1990) Matrix decomposition is complete for the average case. In IEEE FOCS. 2, 802–811.Google Scholar
[15] Gurevich, Y. (1991) Average case complexity. J. Comput. System Sci. 42 346–398.Google Scholar
[16] Hajnal, A. and Szemerédi, E. (1970) Proof of a conjecture of P. Erdős. In Combinatorial Theory and its Applications, II, North-Holland, pp. 601–623.Google Scholar
[17] Ikeno, S. (1958) A 6-symbol 10-state universal Turing machine. In Proc. Inst. of Electrical Communications, Tokyo. (As cited and described in [29].)Google Scholar
[18] Impagliazzo, R. and Levin, L. A. (1990) No better ways to generate hard NP instances than picking uniformly at random. In FOCS '90: 31st Annual Symposium on Foundations of Computer Science, IEEE, pp. 812–821.Google Scholar
[19] Jao, D., Miller, S. D. and Venkatesan, R. (2009) Expander graphs based on GRH with an application to elliptic curve cryptography. J. Number Theory 129 1491–1504.Google Scholar
[20] Karp, R. (1976) The probabilistic analysis of some combinatorial search algorithms. In Algorithms and Complexity (Traub, J. F., ed.), Academic Press, pp. 1–19.Google Scholar
[21] Karp, R., Lenstra, J. K., McDiarmid, C. J. H. and Rinnoy Kan, A. H. G. (1985) Probabilistic analysis. In Combinatorial Optimization: Annotated Bibliographies (O'hEigeartaigh, M., Lenstra, J. K. and Rinnoy Kan, A. H. G., eds), Wiley.Google Scholar
[22] Kierstead, H. A. and Kostochka, A. V. (2008) A short proof of the Hajnal–Szemerédi theorem on equitable colouring. Combin. Probab. Comput. 17 265–270.Google Scholar
[23] Johnson, D. (1984) The NP-completeness column: An ongoing guide. J. Alg. 5 284–299.Google Scholar
[24] Lagarias, J. C. and Odlyzko, A. M. (1983) Solving low density subset sum problems. In 24th Annual Symposium on Foundations of Computer Science, IEEE, pp. 1–10.Google Scholar
[25] Lenstra, A. K. and Lenstra, H. W. (1991) The Development of the Number Field Sieve, Springer.Google Scholar
[26] Levin, L. A. (1986) Average case complete problems. SIAM J. Comput. 15 285–286.Google Scholar
[27] Levin, L. A. (2003) The tale of one-way functions. Prob. Inform. Transm. 39 92–103.Google Scholar
[28] Micciancio, D. and Regev, O. (2004) Worst case to average case reductions using Gaussian measures. In FOCS '04: Annual IEEE Symposium on Foundations of Computer Science, IEEE, pp. 372–381.Google Scholar
[29] Minsky, M. L. (1967) Computation: Finite and Infinite Machines, Prentice Hall.Google Scholar
[30] Shamir, A. (1982) A polynomial algorithm for breaking the basic Merkle–Hellman cryptosystem. In FOCS '82: 23rd Annual Symposium on Foundations of Computer Science, IEEE, pp. 145–152.Google Scholar
[31] Venkatesan, R. and Levin, L. A. (1988) Random instances of a graph coloring problem are hard. In STOC '88: 20th Annual ACM Symposium on Theory of Computing, ACM, pp. 217–222.Google Scholar
[32] Venkatesan, R. and Rajagopalan, S. (1992) Average case intractability of matrix and diophantine problems. In STOC '92: 42nd Annual ACM symposium on Theory of Computing, ACM, pp. 632–642.Google Scholar
[33] Yao, A. C. (1982) Theory and applications of trapdoor functions. In FOCS '82: 23rd Annual Symposium on Foundations of Computer Science, IEEE, pp. 80–91.Google Scholar
[34] Wang, J. (1995) Average case completeness of a word problem for groups. In STOC '95: 27th Annual ACM symposium on Theory of Computing, ACM, pp. 325–334.Google Scholar