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13
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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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.
Submodular Approximation: Samplingbased Algorithms and Lower Bounds
, 2008
"... We introduce several generalizations of classical computer science problems obtained by replacing simpler objective functions with general submodular functions. The new problems include submodular load balancing, which generalizes load balancing or minimummakespan scheduling, submodular sparsest cu ..."
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Cited by 26 (0 self)
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We introduce several generalizations of classical computer science problems obtained by replacing simpler objective functions with general submodular functions. The new problems include submodular load balancing, which generalizes load balancing or minimummakespan scheduling, submodular sparsest cut and submodular balanced cut, which generalize their respective graph cut problems, as well as submodular function minimization with a cardinality lower bound. We establish upper and lower bounds for the approximability of these problems with a polynomial number of queries to a functionvalue oracle. The approximation guarantees for most of our algorithms are of the order of √ n/lnn. We show that this is the inherent difficulty of the problems by proving matching lower bounds. We also give an improved lower bound for the problem of approximately learning a monotone submodular function. In addition, we present an algorithm for approximately learning submodular functions with special structure, whose guarantee is close to the lower bound. Although quite restrictive, the class of functions with this structure includes the ones that are used for lower bounds both by us and in previous work. This demonstrates that if there are significantly stronger lower bounds for this problem, they rely on more general submodular functions.
An algebraic characterisation of complexity for valued constraints
 In: Proceedings CP’06. Volume 4204 of Lecture Notes in Computer Science., SpringerVerlag
, 2006
"... Classical constraint satisfaction is concerned with the feasibility of satisfying a collection of constraints. The extension of this framework to include optimisation is now also being investigated and a theory of socalled soft constraints is being developed. In this extended framework, tuples of ..."
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Cited by 14 (5 self)
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Classical constraint satisfaction is concerned with the feasibility of satisfying a collection of constraints. The extension of this framework to include optimisation is now also being investigated and a theory of socalled soft constraints is being developed. In this extended framework, tuples of values allowed by constraints are given desirability weightings, or costs, and the goal is to find the most desirable (or least cost) assignment. The complexity of any optimisation problem depends critically on the type of function which has to be minimized. For soft constraint problems this function is a sum of cost functions chosen from some fixed set of available cost functions, known as a valued constraint language. We show in this paper that when the costs are rational numbers or infinite the complexity of a soft constraint problem is determined by certain algebraic properties of the valued constraint language, which we call feasibility polymorphisms and fractional polymorphisms. As an immediate application of these results, we show that the existence of a nontrivial fractional polymorphism is a necessary condition for the tractability of a valued constraint language with rational or infinite costs over any finite domain (assuming P ≠ NP).
Soft arc consistency revisited
 Artificial Intelligence
"... The Valued Constraint Satisfaction Problem (VCSP) is a generic optimization problem defined by a network of local cost functions defined over discrete variables. It has applications in Artificial Intelligence, Operations Research, Bioinformatics and has been used to tackle optimization problems in o ..."
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Cited by 9 (3 self)
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The Valued Constraint Satisfaction Problem (VCSP) is a generic optimization problem defined by a network of local cost functions defined over discrete variables. It has applications in Artificial Intelligence, Operations Research, Bioinformatics and has been used to tackle optimization problems in other graphical models (including discrete Markov Random Fields and Bayesian Networks). The incremental lower bounds produced by local consistency filtering are used for pruning inside Branch and Bound search. In this paper, we extend the notion of arc consistency by allowing fractional weights and by allowing several arc consistency operations to be applied simultaneously. Over the rationals and allowing simultaneous operations, we show that an optimal arc consistency closure can theoretically be determined in polynomial time by reduction to linear programming. This defines Optimal Soft Arc Consistency (OSAC). To reach a more practical algorithm, we show that the existence of a sequence of arc consistency operations which increases the lower bound can be detected by establishing arc consistency in a classical Constraint Satisfaction Problem (CSP) derived from the original cost function network. This leads to a new soft arc consistency method, called,Virtual Arc Consistency which produces improved lower bounds compared with previous techniques and which can solve submodular cost functions. These algorithms have been implemented and evaluated on a variety of problems, including two difficult frequency assignment problems which are solved to optimality for the first time. Our implementation is available in the open source toulbar2 platform.
Efficient Algorithms for Robustness in Matroid Optimization
 PROCEEDINGS OF THE EIGHTH ANNUAL ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS (NEW
, 1996
"... The robustness function of a matroid measures the maximum increase in the weight of its minimum weight bases that can be produced by increases of a given total cost on the weights of its elements. We present an algorithm for computing this function, that runs in strongly polynomial time for matroids ..."
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Cited by 8 (1 self)
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The robustness function of a matroid measures the maximum increase in the weight of its minimum weight bases that can be produced by increases of a given total cost on the weights of its elements. We present an algorithm for computing this function, that runs in strongly polynomial time for matroids in which independence can be tested in strongly polynomial time. We identify key properties of transversal, scheduling and partition matroids, and exploit them to design robustness algorithms that are more efficient than our general algorithm.
A Strongly Polynomial Algorithm for Line Search in Submodular Polyhedra
 Proceedings of the 4th JapaneseHungarian Symposium on Discrete Mathematics and Its Applications
, 2005
"... A submodular polyhedron is a polyhedron associated with a submodular function. This paper presents a strongly polynomial time algorithm for line search in submodular polyhedra with the aid of a fully combinatorial algorithm for submodular function minimization. The algorithm is based on the parametr ..."
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Cited by 7 (1 self)
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A submodular polyhedron is a polyhedron associated with a submodular function. This paper presents a strongly polynomial time algorithm for line search in submodular polyhedra with the aid of a fully combinatorial algorithm for submodular function minimization. The algorithm is based on the parametric search method proposed by Megiddo. 1
Graph Based Algorithms for Scene Reconstruction from Two or More Views
, 2004
"... In recent years, graph cuts have emerged as a powerful optimization technique for minimizing energy functions that arise in lowlevel vision problems. Graph cuts avoid the problems of local minima inherent in other approaches (such as gradient descent). The goal of this thesis is to apply graph cuts ..."
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Cited by 6 (1 self)
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In recent years, graph cuts have emerged as a powerful optimization technique for minimizing energy functions that arise in lowlevel vision problems. Graph cuts avoid the problems of local minima inherent in other approaches (such as gradient descent). The goal of this thesis is to apply graph cuts to a classical computer vision problem — scene reconstruction from multiple views, i.e. computing the 3dimensional shape of the scene. This thesis provides a technical result which greatly facilitates the derivation of the scene reconstruction algorithm. Our result should also be useful for developing other energy minimization algorithms based on graph cuts. Previously such algorithms explicitly constructed graphs where a minimum cut also minimizes the appropriate energy. It is natural to ask for what energy functions we can construct such a graph. We answer this question for the class of functions of binary variables that can be written as a sum of terms containing three or fewer variables. We give a simple criterion for functions in this class which is necessary and sufficient, as well as a necessary condition for any function of binary variables. We also give a
Dynamic Programming and Graph Algorithms in Computer Vision
"... Optimization is a powerful paradigm for expressing and solving problems in a wide range of areas, and has been successfully applied to many vision problems. Discrete optimization techniques are especially interesting, since by carefully exploiting problem structure they often provide nontrivial gua ..."
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Cited by 6 (0 self)
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Optimization is a powerful paradigm for expressing and solving problems in a wide range of areas, and has been successfully applied to many vision problems. Discrete optimization techniques are especially interesting, since by carefully exploiting problem structure they often provide nontrivial guarantees concerning solution quality. In this paper we briefly review dynamic programming and graph algorithms, and discuss representative examples of how these discrete optimization techniques have been applied to some classical vision problems. We focus on the lowlevel vision problem of stereo; the midlevel problem of interactive object segmentation; and the highlevel problem of modelbased recognition.
Recent Progress in Submodular Function Minimization
, 2000
"... This article is an attempt to relate the history and importance of the problem, the difficulties that arose in confronting it, the motivation for the solutions proposed, some consequences of the existence of these new algorithms, and some further challenges ..."
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Cited by 5 (0 self)
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This article is an attempt to relate the history and importance of the problem, the difficulties that arose in confronting it, the motivation for the solutions proposed, some consequences of the existence of these new algorithms, and some further challenges