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49
Fast approximate energy minimization via graph cuts
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2001
"... In this paper we address the problem of minimizing a large class of energy functions that occur in early vision. The major restriction is that the energy function’s smoothness term must only involve pairs of pixels. We propose two algorithms that use graph cuts to compute a local minimum even when v ..."
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Cited by 1384 (51 self)
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In this paper we address the problem of minimizing a large class of energy functions that occur in early vision. The major restriction is that the energy function’s smoothness term must only involve pairs of pixels. We propose two algorithms that use graph cuts to compute a local minimum even when very large moves are allowed. The first move we consider is an αβswap: for a pair of labels α, β, this move exchanges the labels between an arbitrary set of pixels labeled α and another arbitrary set labeled β. Our first algorithm generates a labeling such that there is no swap move that decreases the energy. The second move we consider is an αexpansion: for a label α, this move assigns an arbitrary set of pixels the label α. Our second
Proof verification and hardness of approximation problems
 IN PROC. 33RD ANN. IEEE SYMP. ON FOUND. OF COMP. SCI
, 1992
"... We show that every language in NP has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof. If a string is in the language, then there exists a proof such that the verifier accepts with probabilit ..."
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Cited by 719 (46 self)
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We show that every language in NP has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof. If a string is in the language, then there exists a proof such that the verifier accepts with probability 1 (i.e., for every choice of its random string). For strings not in the language, the verifier rejects every provided “proof " with probability at least 1/2. Our result builds upon and improves a recent result of Arora and Safra [6] whose verifiers examine a nonconstant number of bits in the proof (though this number is a very slowly growing function of the input length). As a consequence we prove that no MAX SNPhard problem has a polynomial time approximation scheme, unless NP=P. The class MAX SNP was defined by Papadimitriou and Yannakakis [82] and hard problems for this class include vertex cover, maximum satisfiability, maximum cut, metric TSP, Steiner trees and shortest superstring. We also improve upon the clique hardness results of Feige, Goldwasser, Lovász, Safra and Szegedy [42], and Arora and Safra [6] and shows that there exists a positive ɛ such that approximating the maximum clique size in an Nvertex graph to within a factor of N ɛ is NPhard.
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 694 (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.
Global Optimizations for Parallelism and Locality on Scalable Parallel Machines
 IN PROCEEDINGS OF THE SIGPLAN '93 CONFERENCE ON PROGRAMMING LANGUAGE DESIGN AND IMPLEMENTATION
, 1993
"... Data locality is critical to achieving high performance on largescale parallel machines. Nonlocal data accesses result in communication that can greatly impact performance. Thus the mapping, or decomposition, of the computation and data onto the processors of a scalable parallel machine is a key i ..."
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Cited by 240 (21 self)
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Data locality is critical to achieving high performance on largescale parallel machines. Nonlocal data accesses result in communication that can greatly impact performance. Thus the mapping, or decomposition, of the computation and data onto the processors of a scalable parallel machine is a key issue in compiling programs for these architectures.
Markov random fields with efficient approximations
 In IEEE Conference on Computer Vision and Pattern Recognition
, 1998
"... Markov Random Fields (MRF’s) can be used for a wide variety of vision problems. In this paper we focus on MRF’s with twovalued clique potentials, which form a generalized Potts model. We show that the maximum a posteriori estimate of such an MRF can be obtained by solving a multiway minimum cut pro ..."
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Cited by 165 (22 self)
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Markov Random Fields (MRF’s) can be used for a wide variety of vision problems. In this paper we focus on MRF’s with twovalued clique potentials, which form a generalized Potts model. We show that the maximum a posteriori estimate of such an MRF can be obtained by solving a multiway minimum cut problem on a graph. We develop efficient algorithms for computing good approximations to the minimum multiway cut. The visual correspondence problem can be formulated as an MRF in our framework; this yields quite promising results on real data with ground truth. We also apply our techniques to MRF’s with linear clique potentials. 1
Approximate MaxFlow Min(multi)cut Theorems and Their Applications
 SIAM Journal on Computing
, 1993
"... Consider the multicommodity flow problem in which the object is to maximize the sum of commodities routed. We prove the following approximate maxflow minmulticut theorem: min multicut O(logk) max flow min multicut; where k is the number of commodities. Our proof is constructive; it enables us ..."
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Cited by 144 (3 self)
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Consider the multicommodity flow problem in which the object is to maximize the sum of commodities routed. We prove the following approximate maxflow minmulticut theorem: min multicut O(logk) max flow min multicut; where k is the number of commodities. Our proof is constructive; it enables us to find a multicut within O(log k) of the max flow (and hence also the optimal multicut). In addition, the proof technique provides a unified framework in which one can also analyse the case of flows with specified demands, of LeightonRao and Klein et.al., and thereby obtain an improved bound for the latter problem. 1 Introduction Much of flow theory, and the theory of cuts in graphs, is built around a single theorem  the celebrated maxflow mincut theorem of Ford and Fulkerson [FF], and Elias, Feinstein and Shannon [EFS]. The power of this theorem lies in that it relates two fundamental graphtheoretic entities via the potent mechanism of a minmax relation. The importance of this theor...
An O(log k) approximate mincut maxflow theorem and approximation algorithm
 SIAM J. Comput
, 1998
"... Abstract. It is shown that the minimum cut ratio is within a factor of O(log k) of the maximum concurrent flow for kcommodity flow instances with arbitrary capacities and demands. This improves upon the previously bestknown bound of O(log 2 k) and is existentially tight, up to a constant factor. A ..."
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Cited by 126 (6 self)
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Abstract. It is shown that the minimum cut ratio is within a factor of O(log k) of the maximum concurrent flow for kcommodity flow instances with arbitrary capacities and demands. This improves upon the previously bestknown bound of O(log 2 k) and is existentially tight, up to a constant factor. An algorithm for finding a cut with ratio within a factor of O(log k) of the maximum concurrent flow, and thus of the optimal mincut ratio, is presented.
Maximizing Loop Parallelism and Improving Data Locality via Loop Fusion and Distribution
 IN LANGUAGES AND COMPILERS FOR PARALLEL COMPUTING
, 1994
"... Loop fusion is a program transformation that merges multiple loops into one. It is effective for reducing the synchronization overhead of parallel loops and for improving data locality. This paper presents three results for fusion: (1) a new algorithm for fusing a collection of parallel and seq ..."
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Cited by 121 (10 self)
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Loop fusion is a program transformation that merges multiple loops into one. It is effective for reducing the synchronization overhead of parallel loops and for improving data locality. This paper presents three results for fusion: (1) a new algorithm for fusing a collection of parallel and sequential loops, minimizing parallel loop synchronization while maximizing parallelism; (2) a proof that performing fusion to maximize data locality is NPhard; and (3) two polynomialtime algorithms for improving data locality. These techniques also apply to loop distribution, which is shown to be essentially equivalent to loop fusion. Our approach is general enough to support other fusion heuristics. Preliminary experimental results validate our approach for improving performance by exploiting data locality and increasing the granularity of parallelism.
Hardness Of Approximations
, 1996
"... This chapter is a selfcontained survey of recent results about the hardness of approximating NPhard optimization problems. ..."
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Cited by 102 (4 self)
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This chapter is a selfcontained survey of recent results about the hardness of approximating NPhard optimization problems.
A new approach to the minimum cut problem
 Journal of the ACM
, 1996
"... Abstract. This paper presents a new approach to finding minimum cuts in undirected graphs. The fundamental principle is simple: the edges in a graph’s minimum cut form an extremely small fraction of the graph’s edges. Using this idea, we give a randomized, strongly polynomial algorithm that finds th ..."
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Cited by 97 (8 self)
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Abstract. This paper presents a new approach to finding minimum cuts in undirected graphs. The fundamental principle is simple: the edges in a graph’s minimum cut form an extremely small fraction of the graph’s edges. Using this idea, we give a randomized, strongly polynomial algorithm that finds the minimum cut in an arbitrarily weighted undirected graph with high probability. The algorithm runs in O(n 2 log 3 n) time, a significant improvement over the previous Õ(mn) time bounds based on maximum flows. It is simple and intuitive and uses no complex data structures. Our algorithm can be parallelized to run in �� � with n 2 processors; this gives the first proof that the minimum cut problem can be solved in ���. The algorithm does more than find a single minimum cut; it finds all of them. With minor modifications, our algorithm solves two other problems of interest. Our algorithm finds all cuts with value within a multiplicative factor of � of the minimum cut’s in expected Õ(n 2 � ) time, or in �� � with n 2 � processors. The problem of finding a minimum multiway cut of a graph into r pieces is solved in expected Õ(n 2(r�1) ) time, or in �� � with n 2(r�1) processors. The “trace ” of the algorithm’s execution on these two problems forms a new compact data structure for representing all small cuts and all multiway cuts in a graph. This data structure can be efficiently transformed into the