Abstract
The problem of finding the most probable explanation to a designated set of variables (the MAP problem) is a notoriously intractable problem in Bayesian networks, both to compute exactly and to approximate. It is known, both from theoretical considerations and from practical experiences, that low treewidth is typically an essential prerequisite to efficient exact computations in Bayesian networks. In this paper we investigate whether the same holds for approximating MAP. We define four notions of approximating MAP (by value, structure, rank, and expectation) and argue that all of them are intractable in general. We prove that efficient value-, structure-, and rank-approximations of MAP instances with high treewidth will violate the Exponential Time Hypothesis. In contrast, we hint that expectation-approximation can be done efficiently, even in MAP instances with high treewidth, if the most probable explanation has a high probability.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abdelbar, A.M., Hedetniemi, S.M.: Approximating MAPs for belief networks is NP-hard and other theorems. Artificial Intelligence 102, 21–38 (1998)
Agrawal, M., Kayal, N., Saxena, N.: PRIMES is in P. Annals of Mathematics 160(2), 781–793 (2004)
Arora, S., Barak, B.: Computational Complexity: A Modern Approach. Cambridge University Press (2009)
Darwiche, A.: Modeling and Reasoning with Bayesian Networks. Cambridge University Press (2009)
De Campos, C.P.: New complexity results for MAP in Bayesian networks. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence, pp. 2100–2106 (2011)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Berlin (1999)
Hamilton, M., Müller, M., van Rooij, I., Wareham, H.T.: Approximating solution structure. In: Demaine, E., Gutin, G.Z., Marx, D., Stege, U. (eds.) Structure Theory and FPT Algorithmics for Graphs, Digraphs and Hypergraphs. Dagstuhl Seminar Proceedings, vol. (07281) (2007)
Impagliazzo, R., Paturi, R.: On the complexity of k-SAT. Journal of Computer and System Sciences 62(2), 367–375 (2001)
Krentel, M.W.: The complexity of optimization problems. Journal of Computer and System Sciences 36, 490–509 (1988)
Kwisthout, J.: The computational complexity of probabilistic inference. Technical Report ICIS–R11003, Radboud University Nijmegen (2011)
Kwisthout, J.: Most probable explanations in Bayesian networks: Complexity and tractability. International Journal of Approximate Reasoning 52(9), 1452–1469 (2011)
Kwisthout, J.: Structure approximation of most probable explanations in Bayesian networks. In: van der Gaag, L.C. (ed.) ECSQARU 2013. LNCS (LNAI), vol. 7958, pp. 340–351. Springer, Heidelberg (2013)
Kwisthout, J., Bodlaender, H.L., van der Gaag, L.C.: The necessity of bounded treewidth for efficient inference in Bayesian networks. In: Coelho, H., Studer, R., Wooldridge, M. (eds.) Proceedings of the 19th European Conference on Artificial Intelligence (ECAI 2010), pp. 237–242. IOS Press (2010)
Kwisthout, J.H.P., Bodlaender, H.L., van der Gaag, L.C.: The complexity of finding kth most probable explanations in probabilistic networks. In: Černá, I., Gyimóthy, T., Hromkovič, J., Jefferey, K., Králović, R., Vukolić, M., Wolf, S. (eds.) SOFSEM 2011. LNCS, vol. 6543, pp. 356–367. Springer, Heidelberg (2011)
Kwisthout, J., van Rooij, I.: Bridging the gap between theory and practice of approximate Bayesian inference. Cognitive Systems Research 24, 2–8 (2013)
Marx, D.: Can you beat treewidth? In: Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2007), pp. 169–179 (2007)
Park, J.D., Darwiche, A.: Complexity results and approximation settings for MAP explanations. Journal of Artificial Intelligence Research 21, 101–133 (2004)
Robertson, N., Seymour, P.D.: Graph minors II: Algorithmic aspects of tree-width. Journal of Algorithms 7, 309–322 (1986)
Valiant, L.G.: A theory of the learnable. Communications of the ACM 27(11), 1134–1142 (1984)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Kwisthout, J. (2014). Treewidth and the Computational Complexity of MAP Approximations. In: van der Gaag, L.C., Feelders, A.J. (eds) Probabilistic Graphical Models. PGM 2014. Lecture Notes in Computer Science(), vol 8754. Springer, Cham. https://doi.org/10.1007/978-3-319-11433-0_18
Download citation
DOI: https://doi.org/10.1007/978-3-319-11433-0_18
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-11432-3
Online ISBN: 978-3-319-11433-0
eBook Packages: Computer ScienceComputer Science (R0)