Exact Solution of Permuted Submodular MinSum Problems

Schlesinger, Dmitrij

doi:10.1007/978-3-540-74198-5_3

Cited by 32 publications

(32 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For certain subclasses of the problem, it is possible to compute the exact solution in polynomial time: MRFs of bounded tree-width, e.g. (Lauritzen, 1998); with convex pairwise potentials (Ishikawa, 2003); with submodular potentials of binary (Hammer, 1965;Kolmogorov & Zabih, 2004) or multi-label (Schlesinger & Flach, 2000;Kovtun, 2004) variables; with permuted submodular potentials (Schlesinger, 2007). However, the problem of minimizing a general energy function is NPhard.…”

Section: Introductionmentioning

confidence: 99%

On partial optimality in multi-label MRFs

Kohli

Shekhovtsov

Rother

et al. 2008

Proceedings of the 25th International Conference on Machine Learning - ICML '08

View full text Add to dashboard Cite

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 binarylabel 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.

show abstract

Section: Introductionmentioning

confidence: 99%

On partial optimality in multi-label MRFs

Kohli

Shekhovtsov

Rother

et al. 2008

Proceedings of the 25th International Conference on Machine Learning - ICML '08

View full text Add to dashboard Cite

show abstract

“…Unimodular functions are recognizable in polynomial time (Crama, 1989). If the switching operations can be used to make the function supermodular (but not polar) it is a permutable supermodular function (Schlesinger, 2007), which is mainly interesting for the optimization of functions with nonbinary discrete variables.…”

Section: Classes Of Tractable Pseudo-boolean Functionsmentioning

confidence: 99%

Quadratization and Roof Duality of Markov Logic Networks

Nijs

Landsiedel²,

Wollher³

et al. 2016

jair

View full text Add to dashboard Cite

This article discusses the quadratization of Markov Logic Networks, which enables efficient approximate MAP computation by means of maximum flows. The procedure relies on a pseudo-Boolean representation of the model, and allows handling models of any order. The employed pseudo-Boolean representation can be used to identify problems that are guaranteed to be solvable in low polynomial-time. Results on common benchmark problems show that the proposed approach finds optimal assignments for most variables in excellent computational time and approximate solutions that match the quality of ILPbased solvers.

show abstract

“…Moreover, [9], [12] show that to solve a supermodular WCSP it suffices to find any local optimum of the bound such that f ϕ is (G)AC. D. Schlesinger [15] showed that binary supermodular WCSPs can be solved in polynomial time even after an unknown permutation of states in each variable. As pointed out in [15], [12], this can be done also for n-ary supermodular WCSPs.…”

Section: Supermodular Problemsmentioning

confidence: 99%

“…Revisiting [50], [8], [9], [12], [15], we show in this section how to solve permuted n-ary supermodular WCSPs.…”

Section: Supermodular Problemsmentioning

confidence: 99%

Revisiting the Linear Programming Relaxation Approach to Gibbs Energy Minimization and Weighted Constraint Satisfaction

Werner

2010

IEEE Trans. Pattern Anal. Mach. Intell.

View full text Add to dashboard Cite

Abstract-We present a number of contributions to the LP relaxation approach to weighted constraint satisfaction (= Gibbs energy minimization). We link this approach to many works from constraint programming, which relation has so far been ignored in machine vision and learning. While the approach has been mostly considered only for binary constraints, we generalize it to n-ary constraints in a simple and natural way. This includes a simple algorithm to minimize the LP-based upper bound, n-ary max-sum diffusionhowever, we consider using other bound-optimizing algorithms as well. The diffusion iteration is tractable for a certain class of higharity constraints represented as a black-box, which is analogical to propagators for global constraints CSP. Diffusion exactly solves permuted n-ary supermodular problems. A hierarchy of gradually tighter LP relaxations is obtained simply by adding various zero constraints and coupling them in various ways to existing constraints. Zero constraints can be added incrementally, which leads to a cutting plane algorithm. The separation problem is formulated as finding an unsatisfiable subproblem of a CSP.

show abstract

Exact Solution of Permuted Submodular MinSum Problems

Cited by 32 publications

References 9 publications

On partial optimality in multi-label MRFs

On partial optimality in multi-label MRFs

Quadratization and Roof Duality of Markov Logic Networks

Revisiting the Linear Programming Relaxation Approach to Gibbs Energy Minimization and Weighted Constraint Satisfaction

Contact Info

Product

Resources

About