Abstract
An important subclass of the min-sum labeling problem (also known as discrete energy minimization or valued constraint satisfaction) is the pairwise min-sum problem with arbitrary unary costs and attractive Potts pairwise costs (also known as the uniform metric labeling problem). In analogy with our recent result, we show that solving the LP relaxation of the Potts min-sum problem is not significantly easier than that of the general min-sum problem and thus, in turn, the general linear program. This suggests that trying to find an efficient algorithm to solve the LP relaxation of the Potts min-sum problem has a fundamental limitation. Our constructions apply also to integral solutions, yielding novel reductions of the (non-relaxed) general min-sum problem to the Potts min-sum problem.
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References
Boros, E., Hammer, P.L.: Pseudo-Boolean optimization. Discrete Applied Mathematics 123(1-3), 155–225 (2002)
Boykov, Y., Veksler, O., Zabih, R.: Fast approximate energy minimization via graph cuts. IEEE Trans. Pattern Analysis and Machine Intelligence 23(11), 1222–1239 (2001)
Chekuri, C., Khanna, S., Naor, J., Zosin, L.: A linear programming formulation and approximation algorithms for the metric labeling problem. SIAM Journal on Discrete Mathematics 18(3), 608–625 (2005)
Chuzhoy, J., Naor, J.: The hardness of metric labeling. SIAM J. Computation 36(5), 1376–1386 (2007)
Kappes, J.H., Andres, B., Hamprecht, F.A., Schnörr, C., Nowozin, S., Batra, D., Kim, S., Kausler, B.X., Lellmann, J., Komodakis, N., Rother, C.: A comparative study of modern inference techniques for discrete energy minimization problem. In: Conf. on Computer Vision and Pattern Recognition (2013)
Kleinberg, J., Tardos, E.: Approximation algorithms for classification problems with pairwise relationships: Metric labeling and markov random fields. J. ACM 49(5), 616–639 (2002)
Koster, A., van Hoesel, S.P.M., Kolen, A.W.J.: The partial constraint satisfaction problem: Facets and lifting theorems. Operations Research Letters 23(3-5), 89–97 (1998)
Kovtun, I.: Partial optimal labelling search for a NP-hard subclass of (max,+) problems. In: Conf. German Assoc. for Pattern Recognition, pp. 402–409 (2003)
Osokin, A., Vetrov, D., Kolmogorov, V.: Submodular decomposition framework for inference in associative Markov networks with global constraints. In: IEEE Conf. on Computer Vision and Pattern Recognition, pp. 1889–1896 (2011)
Průša, D., Werner, T.: Universality of the local marginal polytope. In: Conf. on Computer Vision and Pattern Recognition, pp. 1738–1743. IEEE Computer Society (2013)
Rother, C., Kolmogorov, V., Lempitsky, V.S., Szummer, M.: Optimizing binary MRFs via extended roof duality. In: Conf. on Computer Vision and Pattern Recognition (2007)
Schlesinger, D., Flach, B.: Transforming an arbitrary MinSum problem into a binary one. Tech. Rep. TUD-FI06-01, Dresden University of Technology, Germany (April 2006)
Shlezinger, M.I.: Syntactic analysis of two-dimensional visual signals in noisy conditions. Cybernetics and Systems Analysis 12(4), 612–628 (1976)
Swoboda, P., Savchynskyy, B., Kappes, J.H., Schnörr, C.: Partial optimality via iterative pruning for the Potts model. In: Conf. on Scale Space and Variational Methods in Computer Vision (2013)
Szeliski, R., Zabih, R., Scharstein, D., Veksler, O., Kolmogorov, V., Agarwala, A., Tappen, M., Rother, C.: A comparative study of energy minimization methods for Markov random fields with smoothness-based priors. IEEE Trans. on Pattern Analysis and Machine Intelligence 30(6), 1068–1080 (2008)
Živný, S.: The Complexity of Valued Constraint Satisfaction Problems. Cognitive Technologies. Springer (2012)
Wainwright, M.J., Jordan, M.I.: Graphical models, exponential families, and variational inference. Foundations and Trends in Machine Learning 1(1-2), 1–305 (2008)
Werner, T.: A linear programming approach to max-sum problem: A review. IEEE Trans. Pattern Analysis and Machine Intelligence 29(7), 1165–1179 (2007)
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Průša, D., Werner, T. (2015). How Hard Is the LP Relaxation of the Potts Min-Sum Labeling Problem?. In: Tai, XC., Bae, E., Chan, T.F., Lysaker, M. (eds) Energy Minimization Methods in Computer Vision and Pattern Recognition. EMMCVPR 2015. Lecture Notes in Computer Science, vol 8932. Springer, Cham. https://doi.org/10.1007/978-3-319-14612-6_5
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DOI: https://doi.org/10.1007/978-3-319-14612-6_5
Publisher Name: Springer, Cham
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