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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 690 (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.
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...
The approximability of threevalued Max CSP
 SIAM Journal on Computing
"... In the maximum constraint satisfaction problem (Max CSP), one is given a finite collection of (possibly weighted) constraints on overlapping sets of variables, and the goal is to assign values from a given domain to the variables so as to maximize the number (or the total weight, for the weighted ca ..."
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Cited by 15 (9 self)
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In the maximum constraint satisfaction problem (Max CSP), one is given a finite collection of (possibly weighted) constraints on overlapping sets of variables, and the goal is to assign values from a given domain to the variables so as to maximize the number (or the total weight, for the weighted case) of satisfied constraints. This problem is NPhard in general, and, therefore, it is natural to study how restricting the allowed types of constraints affects the approximability of the problem. It is known that every Boolean (that is, twovalued) Max CSP problem with a finite set of allowed constraint types is either solvable exactly in polynomial time or else APXcomplete (and hence can have no polynomial time approximation scheme unless P = NP). It has been an open problem for several years whether this result can be extended to nonBoolean Max CSP, which is much more difficult to analyze than the Boolean case. In this paper, we make the first step in this direction by establishing this result for Max CSP over a threeelement domain. Moreover, we present a simple description of all polynomialtime solvable cases of our problem. This description uses the wellknown algebraic combinatorial property of supermodularity. We also show that every hard threevalued Max CSP problem contains, in a certain specified sense, one of the two basic hard Max CSP problems which are the Maximum kcolourable subgraph problems for k = 2, 3.
Structural Aspects Of Ordered Polymatroids
, 1997
"... This paper generalizes some aspects of polymatroid theory to partially ordered sets. The investigations are mainly based on Faigle and Kern, Submodular Linear Programs on Forests, Mathematical Programming 72 (1996). A slightly modified concept of submodularity is introduced. As a consequence the mai ..."
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Cited by 7 (0 self)
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This paper generalizes some aspects of polymatroid theory to partially ordered sets. The investigations are mainly based on Faigle and Kern, Submodular Linear Programs on Forests, Mathematical Programming 72 (1996). A slightly modified concept of submodularity is introduced. As a consequence the main results do not require any assumptions concerning the underlying partially ordered groundset of the polymatroid. The partial orders are not required to be rooted forests. We consider two different basis concepts for ordered polymatroids. These are Core(f), the set of all elements with maximal cardinality and Max(f), the set of all maximal feasible elements. Both concepts are equivalent for unordered polymatroids. The sets Core(f) and Max(f) are completely described by facetinducing inequalities. Furthermore it is shown by an example that Max(f) is in general not a polyhedral set.
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
Adaptive Lexicographic Optimization in MultiClass M/GI/1 Queues
 Mathematics of Operations Research
, 1993
"... this paper we take a di#erent approach. While the choice of cost function for a particular system is often ad hoc, it is more natural to associate an average response time objective with each class and consider its performance relative to the objective. Specically, let # # be the response time objec ..."
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Cited by 5 (2 self)
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this paper we take a di#erent approach. While the choice of cost function for a particular system is often ad hoc, it is more natural to associate an average response time objective with each class and consider its performance relative to the objective. Specically, let # # be the response time objective and let # denote the longrun average response time (assuming it exists) of class # customers under a scheduling policy #. Attention is restricted to the class # of nonidling, nonpreemptive, and nonanticipative policies; the last term means that scheduling decisions do not depend on future arrival and service times. We are interested in determining a policy in # which lexicographically minimizes the vector of performance ratios ( 1 ## 1 ##### # ## # ) # (1) arranged in nonincreasing order (see Section 2 for the denition.) We will refer to this minimization as lexicographic. Results on optimality crucially hinge on the possibility of characterizing the subset of ## that consists of the vectors of mean response times achievable by policies in #.ThesetA is known to be the base of a polymatroid and is described in Section 2. The lexicographic minimization of vector (1)overthesetA yields a unique point # # := (# # 1 ## 1 ###### # # ## # ) .Suchapointhas certain properties that capture fairness in resource allocation. These are described in remarks following Problem (P) of Section 2. Lexicographic minimization has been studied extensively in a deterministic context [15], [23]. The main contribution of this paper is two simple adaptive policies that (exactly and approximately, respectively) minimize (1) lexicographically. Three quantities are needed in order to specify our policies. Set # 0 =0and denote by # # the end of the #th busy period, # =1# 2##...
Problems of Adaptive Optimization in Multiclass M/GI/1 Queues with Bernoulli Feedback
 Mathematics of Operations Research
, 1992
"... Adaptive algorithms are obtained for the solution of separable optimization problems in multiclass #####1 queues with Bernoulli feedback. Optimality of the algorithms is established by modifying and extending methods of stochastic approximation. These algorithms, can be used as a basis for designing ..."
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Cited by 4 (2 self)
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Adaptive algorithms are obtained for the solution of separable optimization problems in multiclass #####1 queues with Bernoulli feedback. Optimality of the algorithms is established by modifying and extending methods of stochastic approximation. These algorithms, can be used as a basis for designing policies for semiseparable and approximate lexicographic optimization problems and in the case of #####1 queues without feedback, they also provide a simple policy for lexicographic optimization. The results obtained on stochastic approximation imply convergence of classical recursions such as RobbinsMonroe in cases where the conditional second moment of their increments is not nite.
Supermodular Functions and the Complexity of Max CSP
, 2004
"... In this paper we study the complexity of the maximum constraint satisfaction problem (Max CSP) over an arbitrary finite domain. An instance of Max CSP consists of a set of variables and a collection of constraints which are applied to certain specified subsets of these variables; the goal is to find ..."
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In this paper we study the complexity of the maximum constraint satisfaction problem (Max CSP) over an arbitrary finite domain. An instance of Max CSP consists of a set of variables and a collection of constraints which are applied to certain specified subsets of these variables; the goal is to find values for the variables which maximize the number of simultaneously satisfied constraints. Using the theory of sub and supermodular functions on finite latticeordered sets, we obtain the first examples of general families of efficiently solvable cases of Max CSP for arbitrary finite domains. In addition, we provide the first dichotomy result for a special class of nonBoolean Max CSP, by considering binary constraints given by supermodular functions on a totally ordered set. Finally, we show that the equality constraint over a nonBoolean domain is nonsupermodular, and, when combined with some simple unary constraints, gives rise to cases of Max CSP which are hard even to approximate.
AND SUBLATTICES OF R n
, 1298
"... Abstract. A bimonotone linear inequality is a linear inequality with at most two nonzero coefficients that are of opposite signs (if both different from zero). A linear inequality defines a halfspace that is a sublattice of Rn (a subset closed with respect to componentwise maximum and minimum) if an ..."
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Abstract. A bimonotone linear inequality is a linear inequality with at most two nonzero coefficients that are of opposite signs (if both different from zero). A linear inequality defines a halfspace that is a sublattice of Rn (a subset closed with respect to componentwise maximum and minimum) if and only if it is bimonotone. Veinott has shown that a polyhedron is a sublattice if and only if it can be defined by a finite system of bimonotone linear inequalities, whereas Topkis has shown that every sublattice of Rn (and of more general product lattices) is the solution set of a system of nonlinear bimonotone inequalities. In this paper we prove that a subset of Rn is the solution set of a countable system of bimonotone linear inequalities if and only if it is a closed convex sublattice. We also present necessary and/or sufficient conditions for a sublattice to be the intersection of the cartesian product of its projections on the coordinate axes with the solution set of a (possibly infinite) system of bimonotone linear inequalities. We provide explicit constructions of such systems of bimonotone linear inequalities under certain assumptions on the sublattice. We obtain Veinott’s polyhedral representation theorem and a 01 version of Birkhoff’s Representation Theorem as corollaries. We also point out a few potential pitfalls regarding properties of sublattices of Rn. 1.