@article {Shekhovtsov2018,
	title = {Maximum Persistency via Iterative Relaxed Inference in Graphical Models},
	journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence},
	volume = {40},
	number = {7},
	year = {2018},
	pages = {1668{\textendash}1682},
	abstract = {We consider the NP-hard problem of MAP-inference for undirected discrete graphical models. We propose a polynomial time and practically efficient algorithm for finding a part of its optimal solution. Specifically, our algorithm marks some labels of the considered graphical model either as (i) optimal, meaning that they belong to all optimal solutions of the inference problem; (ii) non-optimal if they provably do not belong to any solution. With access to an exact solver of a linear programming relaxation to the MAP-inference problem, our algorithm marks the maximal possible (in a specified sense) number of labels. We also present a version of the algorithm, which has access to a suboptimal dual solver only and still can ensure the (non-)optimality for the marked labels, although the overall number of the marked labels may decrease. We propose an efficient implementation, which runs in time comparable to a single run of a suboptimal dual solver. Our method is well-scalable and shows state-of-the-art results on computational benchmarks from machine learning and computer vision.},
	keywords = {discrete optimization, energy minimization, graphical models, LP relaxation, partial optimality, persistency, WCSP},
	issn = {01628828},
	doi = {10.1109/TPAMI.2017.2730884},
	url = {http://www.icg.tugraz.at/},
	author = {Shekhovtsov, Alexander and Swoboda, Paul and Savchynskyy, Bogdan}
}
@article {Swoboda2016,
	title = {Partial Optimality by Pruning for MAP-Inference with General Graphical Models},
	journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence},
	volume = {38},
	number = {7},
	year = {2016},
	month = {jul},
	pages = {1370{\textendash}1382},
	publisher = {IEEE Computer Society},
	abstract = {We consider the energy minimization problem for undirected graphical models, also known as MAP-inference problem for Markov random fields which is NP-hard in general. We propose a novel polynomial time algorithm to obtain a part of its optimal non-relaxed integral solution. Our algorithm is initialized with variables taking integral values in the solution of a convex relaxation of the MAP-inference problem and iteratively prunes those, which do not satisfy our criterion for partial optimality. We show that our pruning strategy is in a certain sense theoretically optimal. Also empirically our method outperforms previous approaches in terms of the number of persistently labelled variables. The method is very general, as it is applicable to models with arbitrary factors of an arbitrary order and can employ any solver for the considered relaxed problem. Our method{\textquoteright}s runtime is determined by the runtime of the convex relaxation solver for the MAP-inference problem.},
	keywords = {energy minimization, Local polytope, MAP-inference, Markov random fields, partial optimality, persistency},
	issn = {01628828},
	doi = {10.1109/TPAMI.2015.2484327},
	author = {Swoboda, Paul and Shekhovtsov, Alexander and Kappes, Jorg Hendrik and Christoph Schn{\"o}rr and Savchynskyy, Bogdan}
}
@booklet {Kolmogorov2007,
	title = {Minimizing nonsubmodular functions with graph cuts - A review},
	howpublished = {IEEE Transactions on Pattern Analysis and Machine Intelligence},
	volume = {29},
	number = {7},
	year = {2007},
	pages = {1274{\textendash}1279},
	abstract = {Optimization techniques based on graph cuts have become a standard tool for many vision applications. These techniques allow to minimize efficiently certain energy functions corresponding to pairwise Markov Random Fields (MRFs). Currently, there is an accepted view within the computer vision community that graph cuts can only be used for optimizing a limited class of MRF energies (e.g., submodular functions). In this survey, we review some results that show that graph cuts can be applied to a much larger class of energy functions (in particular, nonsubmodular functions). While these results are well-known in the optimization community, to our knowledge they were not used in the context of computer vision and MRF optimization. We demonstrate the relevance of these results to vision on the problem of binary texture restoration. {\textcopyright} 2007 IEEE.},
	keywords = {energy minimization, Markov random fields, Min cut/max flow, Quadratic pseudo-Boolean optimization, Texture restoration},
	issn = {01628828},
	doi = {10.1109/TPAMI.2007.1031},
	author = {Kolmogorov, Vladimir and Carsten Rother}
}
