@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}
}
@conference {Kohli2008,
	title = {On partial optimality in multi-label MRFs},
	booktitle = {Proceedings of the 25th International Conference on Machine Learning},
	year = {2008},
	pages = {480{\textendash}487},
	abstract = {We consider the problem of optimizing multilabel MRFs, which is in general NP-hard and ubiquitous in low-level computer vision. One approach for its solution is to formulate it as an integer linear programming and relax the integrality constraints. The approach we consider in this paper is to first convert the multi-label MRF into an equivalent binary-label MRF and then to relax it. The resulting relaxation can be efficiently solved using a maximum flow algorithm. Its solution provides us with a partially optimal labelling of the binary variables. This partial labelling is then easily transferred to the multi-label problem. We study the theoretical properties of the new relaxation and compare it with the standard one. Specifically, we compare tightness, and characterize a subclass of problems where the two relaxations coincide. We propose several combined algorithms based on the technique and demonstrate their performance on challenging computer vision problems. Copyright 2008 by the author(s)/owner(s).},
	isbn = {9781605582054},
	doi = {10.1145/1390156.1390217},
	author = {Kohli, Pushmeet and Shekhovtsov, Alexander and Carsten Rother and Kolmogorov, Vladimir and Torr, Philip}
}
