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What energy functions can be minimized via graph cuts
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2004
"... Abstract—In the last few years, several new algorithms based on graph cuts have been developed to solve energy minimization problems in computer vision. Each of these techniques constructs a graph such that the minimum cut on the graph also minimizes the energy. Yet, because these graph construction ..."
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Cited by 699 (21 self)
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Abstract—In the last few years, several new algorithms based on graph cuts have been developed to solve energy minimization problems in computer vision. Each of these techniques constructs a graph such that the minimum cut on the graph also minimizes the energy. Yet, because these graph constructions are complex and highly specific to a particular energy function, graph cuts have seen limited application to date. In this paper, we give a characterization of the energy functions that can be minimized by graph cuts. Our results are restricted to functions of binary variables. However, our work generalizes many previous constructions and is easily applicable to vision problems that involve large numbers of labels, such as stereo, motion, image restoration, and scene reconstruction. We give a precise characterization of what energy functions can be minimized using graph cuts, among the energy functions that can be written as a sum of terms containing three or fewer binary variables. We also provide a generalpurpose construction to minimize such an energy function. Finally, we give a necessary condition for any energy function of binary variables to be minimized by graph cuts. Researchers who are considering the use of graph cuts to optimize a particular energy function can use our results to determine if this is possible and then follow our construction to create the appropriate graph. A software implementation is freely available.
Reverse Search for Enumeration
 Discrete Applied Mathematics
, 1993
"... The reverse search technique has been recently introduced by the authors for efficient enumeration of vertices of polyhedra and arrangements. In this paper, we develop this idea in a general framework and show its broader applications to various problems in operations research, combinatorics, and ..."
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Cited by 153 (25 self)
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The reverse search technique has been recently introduced by the authors for efficient enumeration of vertices of polyhedra and arrangements. In this paper, we develop this idea in a general framework and show its broader applications to various problems in operations research, combinatorics, and geometry. In particular, we propose new algorithms for listing (i) all triangulations of a set of n points in the plane, (ii) all cells in a hyperplane arrangement in R d , (iii) all spanning trees of a graph, (iv) all Euclidean (noncrossing) trees spanning a set of n points in the plane, (v) all connected induced subgraphs of a graph, and (vi) all topological orderings of an acyclic graph. Finally we propose a new algorithm for the 01 integer programming problem which can be considered as an alternative to the branchandbound algorithm. 1 Introduction The listing of all objects that satisfy a specified property is a fundamental problem in combinatorics, computational geometr...
Robust Higher Order Potentials for Enforcing Label Consistency
, 2009
"... This paper proposes a novel framework for labelling problems which is able to combine multiple segmentations in a principled manner. Our method is based on higher order conditional random fields and uses potentials defined on sets of pixels (image segments) generated using unsupervised segmentation ..."
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Cited by 137 (24 self)
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This paper proposes a novel framework for labelling problems which is able to combine multiple segmentations in a principled manner. Our method is based on higher order conditional random fields and uses potentials defined on sets of pixels (image segments) generated using unsupervised segmentation algorithms. These potentials enforce label consistency in image regions and can be seen as a generalization of the commonly used pairwise contrast sensitive smoothness potentials. The higher order potential functions used in our framework take the form of the Robust P n model and are more general than the P n Potts model recently proposed by Kohli et al. We prove that the optimal swap and expansion moves for energy functions composed of these potentials can be computed by solving a stmincut problem. This enables the use of powerful graph cut based move making algorithms for performing inference in the framework. We test our method on the problem of multiclass object segmentation by augmenting the conventional CRF used for object segmentation with higher order potentials defined on image regions. Experiments on challenging data sets show that integration of higher order potentials quantitatively and qualitatively improves results leading to much better definition of object boundaries. We
Algorithms in Discrete Convex Analysis
 Math. Programming
, 2000
"... this paper is to describe the f#eA damental results on M and Lconvex f#24L2A+ with special emphasis on algorithmic aspects. ..."
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Cited by 96 (21 self)
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this paper is to describe the f#eA damental results on M and Lconvex f#24L2A+ with special emphasis on algorithmic aspects.
A Combinatorial, Strongly PolynomialTime Algorithm for Minimizing Submodular Functions
, 2000
"... algorithm for minimizing submodular functions, answering an open question posed in 1981 by GrStschel, Lovsz, and Schrijver. The algorithm employs a scaling scheme that uses a flow in the complete directed graph on the underlying set with each arc capacity equal to the scaled parameter. The resulting ..."
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Cited by 60 (5 self)
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algorithm for minimizing submodular functions, answering an open question posed in 1981 by GrStschel, Lovsz, and Schrijver. The algorithm employs a scaling scheme that uses a flow in the complete directed graph on the underlying set with each arc capacity equal to the scaled parameter. The resulting algorithm runs in time bounded by a polynomial in the size of the underlying set and the largest length of the function value. The paper also presents a strongly polynomialtime version that runs in time bounded by a polynomial in the size of the underlying set independent of the function value.
Minimal EdgeCoverings of Pairs of Sets
, 1995
"... A new minmax theorem concerning bisupermodular functions on pairs of sets is proved. As a special case, we derive an extension of (A. Lubiw's extension of) E. Györi's theorem on intervals, W. Mader's theorem on splitting off edges in directed graphs, J. Edmonds' theorem on matroid partitions, and ..."
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Cited by 59 (13 self)
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A new minmax theorem concerning bisupermodular functions on pairs of sets is proved. As a special case, we derive an extension of (A. Lubiw's extension of) E. Györi's theorem on intervals, W. Mader's theorem on splitting off edges in directed graphs, J. Edmonds' theorem on matroid partitions, and an earlier result of the first author on the minimum number of new directed edges whose addition makes a digraph kedgeconnected. As another consequence, we solve the corresponding nodeconnectivity augmentation problem in directed graphs.
A fully combinatorial algorithm for submodular function minimization
 J. COMBIN. THEORY
"... This paper presents a new simple algorithm for minimizing submodular functions. For integer valued submodular functions, the algorithm runs in O(n6EO log nM) time, where n is the cardinality of the ground set, M is the maximum absolute value of the function value, and EO is the time for function eva ..."
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Cited by 45 (5 self)
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This paper presents a new simple algorithm for minimizing submodular functions. For integer valued submodular functions, the algorithm runs in O(n6EO log nM) time, where n is the cardinality of the ground set, M is the maximum absolute value of the function value, and EO is the time for function evaluation. The algorithm can be improved to run in O((n4EO+n 5) log nM) time. The strongly polynomial version of this faster algorithm runs in O((n5EO + n6) log n) time for real valued general submodular functions. These are comparable to the best known running time bounds for submodular function minimization. The algorithm can also be implemented in strongly polynomial time using only additions, subtractions, comparisons, and the oracle calls for function evaluation. This is the first fully combinatorial submodular function minimization algorithm that does not rely on the scaling method.
Approximation in Stochastic Scheduling: The Power of LPbased Priority Policies
, 1998
"... Devices]: Modes of ComputationOnline computation General Terms: ALGORITHMS, THEORY Additional Key Words and Phrases: Stochastic scheduling, Approximation, Worstcase performance, Priority policy, LPrelaxation, WSEPT rule, Asymptotic optimality This research was partially supported by the German ..."
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Cited by 37 (4 self)
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Devices]: Modes of ComputationOnline computation General Terms: ALGORITHMS, THEORY Additional Key Words and Phrases: Stochastic scheduling, Approximation, Worstcase performance, Priority policy, LPrelaxation, WSEPT rule, Asymptotic optimality This research was partially supported by the GermanIsraeli Foundation for Scientific Research and Development (G.I.F.) under grant I 246304.02/97. An extended abstract appeared in the Proceedings of the 2nd Int. Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX'99). Authors' addresses: Rolf H. Mohring and Marc Uetz. Technische Universitat Berlin, Fachbereich Mathematik, Sekr. MA 61, Straße des 17. Juni 136, 10623 Berlin, Germany, Email: fmoehring, uetzg@math.tuberlin.de. Andreas S. Schulz. MIT, Sloan School of Management and Operations Research Center, E53361, 30 Wadsworth St, Cambridge, MA 02139, Email: schulz@mit.edu. Permission to make digital or hard copies of part or all of this work for person...
Deltamatroids, Jump Systems and Bisubmodular Polyhedra
, 1993
"... We relate an axiomatic generalization of matroids, called a jump system, to polyhedra arising from bisubmodular functions. Unlike the case for usual submodularity, the points of interest are not all the integral points in the relevant polyhedron, but form a subset of them. However, we do show that t ..."
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Cited by 33 (0 self)
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We relate an axiomatic generalization of matroids, called a jump system, to polyhedra arising from bisubmodular functions. Unlike the case for usual submodularity, the points of interest are not all the integral points in the relevant polyhedron, but form a subset of them. However, we do show that the convex hull of the set of points of a jump system is a bisubmodular polyhedron, and that the integral points of an integral bisubmodular polyhedron determine a (special) jump system. We also prove addition and composition theorems for jump systems, which have several applications for deltamatroids and matroids. Copyright (C) by the Society for Industrial and Applied Mathematics, in SIAM Journal on Discrete Mathematics, 8 (1995) pp. 1732. y Partially supported by an N.S.E.R.C. International Scientific Exchange Award at Carleton University z Partially supported by an N.S.E.R.C. of Canada operating grant 1 Introduction Matroids are important as a unifying structure in pure combin...
A FASTER SCALING ALGORITHM FOR MINIMIZING SUBMODULAR FUNCTIONS
, 2001
"... Combinatorial strongly polynomial algorithms for minimizing submodular functions have been developed by Iwata,Fleischer,and Fujishige (IFF) and by Schrijver. The IFF algorithm employs a scaling scheme for submodular functions,whereas Schrijver’s algorithm achieves strongly polynomial bound with the ..."
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Cited by 31 (5 self)
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Combinatorial strongly polynomial algorithms for minimizing submodular functions have been developed by Iwata,Fleischer,and Fujishige (IFF) and by Schrijver. The IFF algorithm employs a scaling scheme for submodular functions,whereas Schrijver’s algorithm achieves strongly polynomial bound with the aid of distance labeling. Subsequently,Fleischer and Iwata have described a push/relabel version of Schrijver’s algorithm to improve its time complexity. This paper combines the scaling scheme with the push/relabel framework to yield a faster combinatorial algorithm for submodular function minimization. The resulting algorithm improves over the previously best known bound by essentially a linear factor in the size of the underlying ground set.