Abstract
As a framework for simple but basic statistical inference problems we introduce the genetic Most Likely Solution problem, the task of finding a most likely solution (MLS in short) for a given problem instance under some given probability model. Although many MLS problems are NP-hard, we propose for these problems, to study their average-case complexity under their assumed probability models. We show three examples of MLS problems, and show that “message passing algorithms” (e.g., belief propagation) work reasonably well for these problems. Some of the technical results of this paper are from the author’s recent work (Watanabe and Yamamoto in Lecture Notes in Computer Science, vol. 4142, pp. 277–282, 2006, and Onsjö and Watanabe in Lecture Notes in Computer Science, vol. 4288, pp. 507–516, 2006).
Similar content being viewed by others
References
Boppana, R.B.: Eigenvalues and graph bisection: an average-case analysis. In: Proc. Symposium on Foundations of Computer Science, pp. 280–285 (1987)
Coja-Oghlan, A.: A spectral heuristic for bisecting random graphs. Random Struct. Algorithms 29(3), 351–398 (2006)
Dubhashi, D., Laura, L., Panconesi, A.: Analysis and experimental evaluation of a simple algorithm for collaborative filtering in planted partition models. In: Proc. FST TCS 2003, pp. 168–182 (2003)
Gallager, R.G.: Low density parity check codes. IRE Trans. Inf. Theory IT-8(21), 21–28 (1962)
Garey, M.R., Johnson, D.S.: Computers and Intractability. Bell Telephone Laboratories, Incorporated (1979)
Jerrum, M., Sorkin, G.: The Metropolis algorithm for graph bisection. Discrete Appl. Math. 82(1–3), 155–175 (1998)
Luby, M., Mitzenmacher, M., Shokrollahi, M., Spielman, D.: Improved low-density parity-check codes using irregular graphs. IEEE Trans. Inf. Theory 47(2), 585–598 (2001)
MacKay, D.: Good error-correcting codes based on very sparse matrices. IEEE Trans. Inf. Theory IT-45(2), 399–431 (1999)
McEliece, R., MacKay, D., Cheng, J.: Turbo decoding as an instance of Pearl’s “Belief Propagation” algorithm. IEEE J. Sel. Areas Comm. 16(2) 1998
McSherry, F.: Spectral partition of random graphs. In: Proc. 40th IEEE Sympos. on Foundations of Computer Science (FOCS’99). IEEE (1999)
Onsjö, M., Watanabe, O.: Simple algorithms for graph partition problems. Research Report C-212. Dept. of Math. and Comput. Sci., Tokyo Inst. of Tech. (2005)
Onsjö, M., Watanabe, O.: A simple message passing algorithm for graph partition problem. In: Proc. 17th Int’l Sympos. on Algorithms and Computation (ISAAC’06). LNCS, vol. 4288, pp. 507–516. Springer, Berlin (2006)
Pearl, J.: Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, San Mateo (1988)
Watanabe, O., Yamamoto, M.: Average-case analysis for the MAX-2SAT problem. In: Proc. 9th Int’l Conference on Theory and Application of Satisfiability Testing (SAT’06). LNCS, vol. 4142, pp. 277–282. Springer, Berlin (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
The preliminary version of this paper has been presented at CiE2007 at Siena.
An erratum to this article can be found at http://dx.doi.org/10.1007/s00224-009-9218-2
Rights and permissions
About this article
Cite this article
Onsjö, M., Watanabe, O. Theory of Computing Systems (TOCS) Submission Version Finding Most Likely Solutions. Theory Comput Syst 45, 926–942 (2009). https://doi.org/10.1007/s00224-009-9179-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00224-009-9179-5