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15
On fast approximate submodular minimization
"... We are motivated by an application to extract a representative subset of machine learning training data and by the poor empirical performance we observe of the popular minimum norm algorithm. In fact, for our application, minimum norm can have a running time of about O(n 7) (O(n 5) oracle calls). We ..."
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Cited by 13 (6 self)
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We are motivated by an application to extract a representative subset of machine learning training data and by the poor empirical performance we observe of the popular minimum norm algorithm. In fact, for our application, minimum norm can have a running time of about O(n 7) (O(n 5) oracle calls). We therefore propose a fast approximate method to minimize arbitrary submodular functions. For a large subclass of submodular functions, the algorithm is exact. Other submodular functions are iteratively approximated by tight submodular upper bounds, and then repeatedly optimized. We show theoretical properties, and empirical results suggest significant speedups over minimum norm while retaining higher accuracies.
An algebraic theory of complexity for valued constraints: Establishing a Galois connection
 In MFCS
, 2011
"... Abstract. The complexity of any optimisation problem depends critically on the form of the objective function. Valued constraint satisfaction problems are discrete optimisation problems where the function to be minimised is given as a sum of cost functions defined on specified subsets of variables. ..."
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Abstract. The complexity of any optimisation problem depends critically on the form of the objective function. Valued constraint satisfaction problems are discrete optimisation problems where the function to be minimised is given as a sum of cost functions defined on specified subsets of variables. These cost functions are 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 nonnegative rational numbers or infinite, then the complexity of a valued constraint problem is determined by certain algebraic properties of this valued constraint language, which we call weighted polymorphisms. We define a Galois connection between valued constraint languages and sets of weighted polymorphisms and show how the closed sets of this Galois connection can be characterised. These results provide a new approach in the search for tractable valued constraint languages.
Minimizing a sum of submodular functions
 In Discrete Applied Mathematics
, 2012
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Logsupermodular functions, functional clones and counting CSPs
, 2012
"... Motivated by a desire to understand the computational complexity of counting constraint satisfaction problems (counting CSPs), particularly the complexity of approximation, we study functional clones of functions on the Boolean domain, which are analogous to the familiar relational clones constituti ..."
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Motivated by a desire to understand the computational complexity of counting constraint satisfaction problems (counting CSPs), particularly the complexity of approximation, we study functional clones of functions on the Boolean domain, which are analogous to the familiar relational clones constituting Post’s lattice. One of these clones is the collection of logsupermodular (lsm) functions, which turns out to play a significant role in classifying counting CSPs. In our study, we assume that nonnegative unary functions (weights) are available. Given this, we prove that there are no functional clones lying strictly between the clone of lsm functions and the total clone (containing all functions). Thus, any counting CSP that contains a single nontrivial nonlsm function is computationally as hard as any problem in #P. Furthermore, any nontrivial functional clone (in a sense that will be made precise below) contains the binary function “implies”. As a consequence, all nontrivial counting CSPs (with nonnegative unary weights assumed to be available) are computationally at least as difficult as #BIS, the problem of counting independent sets in a bipartite graph. There is empirical evidence that #BIS is hard to solve, even approximately.
The complexity of valued constraint satisfaction
 BULLETIN OF THE EATCS
, 2014
"... We survey recent results on the broad family of problems that can be cast as valued constraint satisfaction problems. We discuss general methods for analysing the complexity of such problems, give examples of tractable cases, and identify general features of the complexity landscape. ..."
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We survey recent results on the broad family of problems that can be cast as valued constraint satisfaction problems. We discuss general methods for analysing the complexity of such problems, give examples of tractable cases, and identify general features of the complexity landscape.
Cooperative cuts for image segmentation
, 2010
"... We propose a novel framework for graphbased cooperative regularization that uses submodular costs on graph edges. We introduce an efficient iterative algorithm to solve the resulting hard discrete optimization problem, and show that it has a guaranteed approximation factor. The edgesubmodular form ..."
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We propose a novel framework for graphbased cooperative regularization that uses submodular costs on graph edges. We introduce an efficient iterative algorithm to solve the resulting hard discrete optimization problem, and show that it has a guaranteed approximation factor. The edgesubmodular formulation is amenable to the same extensions as standard graph cut approaches, and applicable to a range of problems. We apply this method to the image segmentation problem. Specifically, Here, we apply it to introduce a discount for homogeneous boundaries in binary image segmentation on very difficult images, precisely, long thin objects and color and grayscale images with a shading gradient. The experiments show that significant portions of previously truncated objects are now preserved.
A note on some collapse results of valued constraints
, 2009
"... Valued constraint satisfaction problem (VCSP) is an optimisation framework originally coming from Artificial Intelligence and generalising the classical constraint satisfaction problem (CSP). The VCSP is powerful enough to describe many important classes of problems. In order to investigate the comp ..."
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Valued constraint satisfaction problem (VCSP) is an optimisation framework originally coming from Artificial Intelligence and generalising the classical constraint satisfaction problem (CSP). The VCSP is powerful enough to describe many important classes of problems. In order to investigate the complexity and expressive power of valued constraints, a number of algebraic tools have been developed in the literature. In this note we present alternative proofs of some known results without using the algebraic approach, but by representing valued constraints explicitly by combinations of other valued constraints.
Efficient Minimization of Higher Order Submodular functions using Monotonic Boolean Functions
, 2011
"... Submodular function minimization is a key problem in a wide variety of applications in machine learning, economics, game theory, computer vision and many others. The general solver has a complexity of O(n6 + n5L) where L is the time required to evaluate the function and n is the number of variables ..."
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Submodular function minimization is a key problem in a wide variety of applications in machine learning, economics, game theory, computer vision and many others. The general solver has a complexity of O(n6 + n5L) where L is the time required to evaluate the function and n is the number of variables [22]. On the other hand, many useful applications in computer vision and machine learning applications are defined over a special subclasses of submodular functions in which that can be written as the sum of many submodular cost functions defined over cliques containing few variables. In such functions, the pseudoBoolean (or polynomial) representation [2] of these subclasses are of degree (or order, or clique size) k where k muchlessthan n. In this work, we develop efficient algorithms for the minimization of this useful subclass of submodular functions. To do this, we define novel mapping that transform submodular functions of order k into quadratic ones, which can be efficiently minimized in O(n3) time using a maxflow algorithm. The underlying idea is to use auxiliary variables to model the higher order terms and the transformation is found using a carefully constructed linear program. In particular, we model the auxiliary variables as monotonic Boolean functions, allowing us to obtain a compact transformation using as few auxiliary variables as possible. Specifically, we show that our approach for fourth order function requires only 2 auxiliary variables in contrast to 30 or more variables used in existing approaches. In the general case, we give an upper bound for the number or auxiliary variables required to transform a function of order k using Dedekind number, which is substantially lower than the existing bound of 22k.
Advances in GraphCut Optimization: MultiSurface . . .
, 2011
"... Computer vision is full of problems that are elegantly expressed in terms of mathematical optimization, or energy minimization. This is particularly true of “lowlevel ” inference problems such as cleaning up noisy signals, clustering and classifying data, or estimating 3D points from images. Energi ..."
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Computer vision is full of problems that are elegantly expressed in terms of mathematical optimization, or energy minimization. This is particularly true of “lowlevel ” inference problems such as cleaning up noisy signals, clustering and classifying data, or estimating 3D points from images. Energies let us state each problem as a clear, precise objective function. Minimizing the correct energy would, hypothetically, yield a good solution to the corresponding problem. Unfortunately, even for lowlevel problems we are confronted by energies that are computationally hard—often NPhard—to minimize. As a consequence, a rather large portion of computer vision research is dedicated to proposing better energies and better algorithms for energies. This dissertation presents work along the same line, specifically new energies and algorithms based on graph cuts. We present three distinct contributions. First we consider biomedical segmentation where the object of interest comprises multiple distinct regions of uncertain shape (e.g. blood vessels, airways, bone tissue). We show that this common yet difficult scenario can be modeled as an
Fast approximate submodular minimization: Extended version
, 2011
"... We are motivated by an application to extract a representative subset of machine learning training data and by the poor empirical performance we observe of the popular minimum norm algorithm. In fact, for our application, minimum norm can have a running time of about O(n 7) (O(n 5) oracle calls). We ..."
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Cited by 1 (1 self)
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We are motivated by an application to extract a representative subset of machine learning training data and by the poor empirical performance we observe of the popular minimum norm algorithm. In fact, for our application, minimum norm can have a running time of about O(n 7) (O(n 5) oracle calls). We therefore propose a fast approximate method to minimize arbitrary submodular functions. For a large subclass of submodular functions, the algorithm is exact. Other submodular functions are iteratively approximated by tight submodular upper bounds, and then repeatedly optimized. We show theoretical properties, and empirical results suggest significant speedups over minimum norm while retaining higher accuracies. 1