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45
Approximation Algorithms for Classification Problems with Pairwise Relationships: Metric Labeling and Markov Random Fields
 IN IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE
, 1999
"... In a traditional classification problem, we wish to assign one of k labels (or classes) to each of n objects, in a way that is consistent with some observed data that we have about the problem. An active line of research in this area is concerned with classification when one has information about pa ..."
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Cited by 195 (2 self)
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In a traditional classification problem, we wish to assign one of k labels (or classes) to each of n objects, in a way that is consistent with some observed data that we have about the problem. An active line of research in this area is concerned with classification when one has information about pairwise relationships among the objects to be classified; this issue is one of the principal motivations for the framework of Markov random fields, and it arises in areas such as image processing, biometry, and document analysis. In its most basic form, this style of analysis seeks a classification that optimizes a combinatorial function consisting of assignment costs  based on the individual choice of label we make for each object  and separation costs  based on the pair of choices we make for two "related" objects. We formulate a general classification problem of this type, the metric labeling problem; we show that it contains as special cases a number of standard classification f...
Approximation Algorithms for the Metric Labeling Problem via a New Linear Programming Formulation
, 2000
"... We consider approximation algorithms for the metric labeling problem. Informally speaking, we are given a weighted graph that specifies relations between pairs of objects drawn from a given set of objects. The goal is to find a minimum cost labeling of these objects where the cost of a labeling is d ..."
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Cited by 77 (1 self)
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We consider approximation algorithms for the metric labeling problem. Informally speaking, we are given a weighted graph that specifies relations between pairs of objects drawn from a given set of objects. The goal is to find a minimum cost labeling of these objects where the cost of a labeling is determined by the pairwise relations between the objects and a distance function on labels; the distance function is assumed to be a metric. Each object also incurs an assignment cost that is label, and vertex dependent. The problem was introduced in a recent paper by Kleinberg and Tardos [19], and captures many classification problems that arise in computer vision and related fields. They gave an O(log k log log k) approximation for the general case where k is the number of labels and a 2approximation for the uniform metric case. More recently, Gupta and Tardos [14] gave a 4approximation for the truncated linear metric, a natural nonuniform metric motivated by practical applications to image restoration and visual correspondence. In this paper we introduce a new natural integer programming formulation and show that the integrality gap of its linear relaxation either matches or improves the ratios known for several cases of the metric labeling problem studied until now, providing a unified approach to solving them. Specifically, we show that the integrality gap of our LP is bounded by O(log k log log k) for general metric and 2 for the uniform metric thus matching the ratios in [19]. We also develop an algorithm based on our LP that achieves a ratio of 2 + p 2 ' 3:414 for the truncated linear metric improving the ratio provided by [14]. Our algorithm uses the fact that the integrality gap of our LP is 1 on a linear metric. We believe that our formulation h...
Approximation algorithms for the 0extension problem
 IN PROCEEDINGS OF THE TWELFTH ANNUAL ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS
, 2001
"... In the 0extension problem, we are given a weighted graph with some nodes marked as terminals and a semimetric on the set of terminals. Our goal is to assign the rest of the nodes to terminals so as to minimize the sum, over all edges, of the product of the edge’s weight and the distance between t ..."
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Cited by 70 (3 self)
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In the 0extension problem, we are given a weighted graph with some nodes marked as terminals and a semimetric on the set of terminals. Our goal is to assign the rest of the nodes to terminals so as to minimize the sum, over all edges, of the product of the edge’s weight and the distance between the terminals to which its endpoints are assigned. This problem generalizes the multiway cut problem of Dahlhaus, Johnson, Papadimitriou, Seymour, and Yannakakis and is closely related to the metric labeling problem introduced by Kleinberg and Tardos. We present approximation algorithms for 0Extension. In arbitrary graphs, we present a O(log k)approximation algorithm, k being the number of terminals. We also give O(1)approximation guarantees for weighted planar graphs. Our results are based on a natural metric relaxation of the problem, previously considered by Karzanov. It is similar in flavor to the linear programming relaxation of Garg, Vazirani, and Yannakakis for the multicut problem and similar to relaxations for other graph partitioning problems. We prove that the integrality ratio of the metric relaxation is at least c √ lg k for a positive c for infinitely many k. Our results improve some of the results of Kleinberg and Tardos and they further our understanding on how to use metric relaxations.
Graphs of some CAT(0) complexes
 Adv. Appl. Math
, 1998
"... In this note, we characterize the graphs (1skeletons) of some piecewise Euclidean simplicial and cubical complexes having nonpositive curvature in the sense of Gromov’s CAT(0) inequality. Each such cell complex K is simply connected and obeys a certain flag condition. It turns out that if, in addit ..."
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Cited by 48 (19 self)
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In this note, we characterize the graphs (1skeletons) of some piecewise Euclidean simplicial and cubical complexes having nonpositive curvature in the sense of Gromov’s CAT(0) inequality. Each such cell complex K is simply connected and obeys a certain flag condition. It turns out that if, in addition, all maximal cells are either regular Euclidean cubes or right Euclidean triangles glued in a special way, then the underlying graph G�K � is either a median graph or a hereditary modular graph without two forbidden induced subgraphs. We also characterize the simplicial complexes arising from bridged graphs, a class of graphs whose metric enjoys one of the basic properties of CAT(0) spaces. Additionally, we show that the graphs of all these complexes and some more general classes of graphs have geodesic combings and bicombings verifying the 1 or 2fellow traveler property. © 2000 Academic Press 1.
Metric graph theory and geometry: a survey
 CONTEMPORARY MATHEMATICS
"... The article surveys structural characterizations of several graph classes defined by distance properties, which have in part a general algebraic flavor and can be interpreted as subdirect decomposition. The graphs we feature in the first place are the median graphs and their various kinds of general ..."
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Cited by 44 (14 self)
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The article surveys structural characterizations of several graph classes defined by distance properties, which have in part a general algebraic flavor and can be interpreted as subdirect decomposition. The graphs we feature in the first place are the median graphs and their various kinds of generalizations, e.g., weakly modular graphs, or fibercomplemented graphs, or l1graphs. Several kinds of l1graphs admit natural geometric realizations as polyhedral complexes. Particular instances of these graphs also occur in other geometric contexts, for example, as dual polar graphs, basis graphs of (even ∆)matroids, tope graphs, lopsided sets, or plane graphs with vertex degrees and face sizes bounded from below. Several other classes of graphs, e.g., Helly graphs (as injective objects), or bridged graphs (generalizing chordal graphs), or treelike graphs such as distancehereditary graphs occur in the investigation of graphs satisfying some basic properties of the distance function, such as the Helly property for balls, or the convexity of balls or of the neighborhoods of convex sets, etc. Operators between graphs or complexes relate some of the
A linear programming formulation and approximation algorithms for the metric labeling problem
 SIAM J. Discrete Math
"... We consider approximation algorithms for the metric labeling problem. This problem was introduced in a paper by Kleinberg and Tardos [J. ACM, 49 (2002), pp. 616–630] and captures many classification problems that arise in computer vision and related fields. They gave an O(log k log log k) approximat ..."
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Cited by 43 (1 self)
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We consider approximation algorithms for the metric labeling problem. This problem was introduced in a paper by Kleinberg and Tardos [J. ACM, 49 (2002), pp. 616–630] and captures many classification problems that arise in computer vision and related fields. They gave an O(log k log log k) approximation for the general case, where k is the number of labels, and a 2approximation for the uniform metric case. (In fact, the bound for general metrics can be improved to O(log k) by the work of Fakcheroenphol, Rao, and Talwar [Proceedings
Approximation algorithms for multicommoditytype problems with guarantees independent of the graph size
 IN: PROCEEDINGS, IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS
, 2009
"... Linial, London and Rabani [3] proved that the mincut maxflow ratio for general maximum concurrent flow problems (when there are k commodities) is O(log k). Here we attempt to derive a more general theory of Steiner cut and flow problems, and we prove bounds that are polylogarithmic in k for a muc ..."
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Cited by 31 (4 self)
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Linial, London and Rabani [3] proved that the mincut maxflow ratio for general maximum concurrent flow problems (when there are k commodities) is O(log k). Here we attempt to derive a more general theory of Steiner cut and flow problems, and we prove bounds that are polylogarithmic in k for a much broader class of multicommodity flow and cut problems. Our structural results are motivated by the meta question: Suppose we are given a poly(log n) approximation algorithm for a flow or cut problem when can we give a poly(log k) approximation algorithm for a generalization of this problem to a Steiner cut or flow problem? Thus we require that these approximation guarantees be independent of the size of the graph, and only depend on the number of commodities (or the number of terminal nodes in a Steiner cut problem). For many natural applications (when k = n o(1) ) this yields much stronger guarantees. We construct vertexsparsifiers that approximately preserve the value of all terminal mincuts. We prove such sparsifiers exist through zerosum games and metric geometry, and we construct such sparsifiers through oblivious routing guarantees. These results let us reduce a broad class of multicommoditytype problems to a uniform case (on k nodes) at the cost of a loss of a poly(log k) in the approximation guarantee. We then give poly(log k) approximation algorithms for a number of problems for which such results were previously unknown, such as requirement cut, lmulticut, oblivious 0extension, and natural Steiner generalizations of oblivious routing, mincut linear arrangement and minimum linear arrangement.
An analysis of convex relaxations for MAP estimation of discrete MRFs
 JOURNAL OF MACHINE LEARNING RESEARCH
, 2008
"... The problem of obtaining the maximum a posteriori estimate of a general discrete Markov random field (i.e., a Markov random field defined using a discrete set of labels) is known to be NPhard. However, due to its central importance in many applications, several approximation algorithms have been pr ..."
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Cited by 27 (1 self)
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The problem of obtaining the maximum a posteriori estimate of a general discrete Markov random field (i.e., a Markov random field defined using a discrete set of labels) is known to be NPhard. However, due to its central importance in many applications, several approximation algorithms have been proposed in the literature. In this paper, we present an analysis of three such algorithms based on convex relaxations: (i) LPS: the linear programming (LP) relaxation proposed by Schlesinger (1976) for a special case and independently in Chekuri et al. (2001), Koster et al. (1998), and Wainwright et al. (2005) for the general case; (ii) QPRL: the quadratic programming (QP) relaxation of Ravikumar and Lafferty (2006); and (iii) SOCPMS: the second order cone programming (SOCP) relaxation first proposed by Muramatsu and Suzuki (2003) for two label problems and later extended by Kumar et al. (2006) for a general label set. We show that the SOCPMS and the QPRL relaxations are equivalent. Furthermore, we prove that despite the flexibility in the form of the constraints/objective function offered by QP and SOCP, the LPS relaxation strictly dominates (i.e., provides a better approximation than) QPRL and SOCPMS. We generalize these results by defining a large class of SOCP (and equivalent QP) relaxations
Approximate Classification via Earthmover Metrics
 In SODA ’04: Proceedings of the fifteenth annual ACMSIAM symposium on Discrete algorithms
, 2004
"... Given a metric space (X, d), a natural distance measure on probability distributions over X is the earthmover metric. We use randomized rounding of earthmover metrics to devise new approximation algorithms for two wellknown classification problems, namely, metric labeling and 0extension. ..."
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Cited by 21 (4 self)
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Given a metric space (X, d), a natural distance measure on probability distributions over X is the earthmover metric. We use randomized rounding of earthmover metrics to devise new approximation algorithms for two wellknown classification problems, namely, metric labeling and 0extension.
The hardness of metric labeling
 IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS’04
, 2004
"... The Metric Labeling problem is an elegant and powerful mathematical model capturing a wide range of classification problems. The input to the problem consists of a set of labels and a weighted graph. Additionally, a metric distance function on the labels is defined, and for each label and each verte ..."
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Cited by 16 (3 self)
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The Metric Labeling problem is an elegant and powerful mathematical model capturing a wide range of classification problems. The input to the problem consists of a set of labels and a weighted graph. Additionally, a metric distance function on the labels is defined, and for each label and each vertex, an assignment cost is given. The goal is to find a minimumcost assignment of the vertices to the labels. The cost of the solution consists of two parts: the assignment costs of the vertices and the separation costs of the edges (each edge pays its weight times the distance between the two labels to which its endpoints are assigned). Due to the simple structure and variety of the applications, the problem and its special cases (with various distance functions on the labels) have recently received much attention. Metric Labeling has a known logarithmic approximation, and it has been an open question for several years whether a constant approximation exists. We refute this possibility and show that no constant approximation can be obtained for the problem unless P=NP, and we also show that the problem ishard to approximate, unless NP has quasipolynomial time algorithms.