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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.
Simplex Partitioning via Exponential Clocks and the Multiway Cut Problem (Extended Abstract)
, 2013
"... The MultiwayCut problem is a fundamental graph partitioning problem in which the objective is to find a minimum weight set of edges disconnecting a given set of special vertices called terminals. This problem is NPhard and there is a well known geometric relaxation in which the graph is embedded i ..."
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The MultiwayCut problem is a fundamental graph partitioning problem in which the objective is to find a minimum weight set of edges disconnecting a given set of special vertices called terminals. This problem is NPhard and there is a well known geometric relaxation in which the graph is embedded into a high dimensional simplex. Rounding a solution to the geometric relaxation is equivalent to partitioning the simplex. We present a novel simplex partitioning algorithm which is based on competing exponential clocks and distortion. Unlike previous methods, it utilizes cuts that are not parallel to the faces of the simplex. Applying this partitioning algorithm to the multiway cut problem, we obtain a simple (4/3)approximation algorithm, thus, improving upon the current best known result. This bound is further pushed to obtain an approximation factor of 1.32388. It is known that under the assumption of the unique games conjecture, the best possible approximation for the MultiwayCut problem can be attained via the geometric relaxation.
Hardness of Approximation for Crossing Number
 Discrete Comput. Geom
"... We show that, if P6=NP, there is a constant c0> 1 such that there is no c0approximation algorithm for the crossing number, even when restricted to 3regular graphs. 1 ..."
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We show that, if P6=NP, there is a constant c0> 1 such that there is no c0approximation algorithm for the crossing number, even when restricted to 3regular graphs. 1
Earth Mover’s Distance based Similarity Search at Scale
"... Earth Mover’s Distance (EMD), as a similarity measure, has received a lot of attention in the fields of multimedia and probabilistic databases, computer vision, image retrieval, machine learning, etc. EMD on multidimensional histograms provides better distinguishability between the objects approxima ..."
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Earth Mover’s Distance (EMD), as a similarity measure, has received a lot of attention in the fields of multimedia and probabilistic databases, computer vision, image retrieval, machine learning, etc. EMD on multidimensional histograms provides better distinguishability between the objects approximated by the histograms (e.g., images), compared to classic measures like Euclidean distance. Despite its usefulness, EMD has a high computational cost; therefore, a number of effective filtering methods have been proposed, to reduce the pairs of histograms for which the exact EMD has to be computed, during similarity search. Still, EMD calculations in the refinement step remain the bottleneck of the whole similarity search process. In this paper, we focus on optimizing the refinement phase of EMDbased similarity search by (i) adapting an efficient mincost flow algorithm (SIA) for EMD computation, (ii) proposing a dynamic distance bound, which can be used to terminate an EMD refinement early, and (iii) proposing a dynamic refinement order for the candidates which, paired with a concurrent EMD refinement strategy, reduces the amount of needless computations. Our proposed techniques are orthogonal to and can be easily integrated with the stateoftheart filtering techniques, reducing the cost of EMDbased similarity queries by orders of magnitude. 1.
Approximability and Mathematical Relaxations
, 2012
"... The thesis ascertains the approximability of classic combinatorial optimization problems using mathematical relaxations. The general flavor of results in the thesis is: a problem P is hard to approximate to a factor better than one obtained from the R relaxation, unless the Unique Games Conjecture i ..."
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The thesis ascertains the approximability of classic combinatorial optimization problems using mathematical relaxations. The general flavor of results in the thesis is: a problem P is hard to approximate to a factor better than one obtained from the R relaxation, unless the Unique Games Conjecture is false. Almost optimal inapproximability is shown for a wide set of problems including Metric Labeling, Max. Acyclic Subgraph, various packing and covering problems. The key new idea in this thesis is in coverting hard instances of relaxations (a.k.a integrality gap instances) into a proof of inapproximability (assuming the UGC). In most cases, the hard instances were discovered prior to this work; our results imply that these hard instances are possibly strong bottlenecks in designing approximation algorithms of better quality for these problems. For ordering problems such as Max. Acyclic Subgraph and Feedback Arc Set, such hard instances were previously unknown. For these problems (see chapter 6), we construct such hard instance by using the reduction designed to prove the inapproximability.
SDP gaps and UGC hardness for multiway cut, . . .
"... The connection between integrality gaps and computational hardness of discrete optimization problems is an intriguing question. In recent years, this connection has prominently figured in several tight UGCbased hardness results. We show in this paper a direct way of turning integrality gaps into ha ..."
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The connection between integrality gaps and computational hardness of discrete optimization problems is an intriguing question. In recent years, this connection has prominently figured in several tight UGCbased hardness results. We show in this paper a direct way of turning integrality gaps into hardness results for several fundamental classification problems. Specifically, we convert linear programming integrality gaps for the Multiway Cut, 0Extension and Metric Labeling problems into UGCbased hardness results. Qualitatively, our result suggests that if the unique games conjecture is true then a linear relaxation of the latter problems studied in several papers (socalled earthmover linear program) yields the best possible approximation. Taking this a step further, we also obtain integrality gaps for a semidefinite programming relaxation matching the integrality gaps of the earthmover linear program. Prior to this work, there was an intriguing possibility of obtaining better approximation factors for labeling problems via semidefinite programming.
SDP gaps and UGC hardness for multiway cut, 0extension and . . .
"... The connection between integrality gaps and computational hardness of discrete optimization problems is an intriguing question. In recent years, this connection has prominently figured in several tight UGCbased hardness results. We show in this paper a direct way of turning integrality gaps into ha ..."
Abstract
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The connection between integrality gaps and computational hardness of discrete optimization problems is an intriguing question. In recent years, this connection has prominently figured in several tight UGCbased hardness results. We show in this paper a direct way of turning integrality gaps into hardness results for several fundamental classification problems. Specifically, we convert linear programming integrality gaps for the Multiway Cut, 0Extension and Metric Labeling problems into UGCbased hardness results. Qualitatively, our result suggests that if the unique games conjecture is true then a linear relaxation of the latter problems studied in several papers (socalled earthmover linear program) yields the best possible approximation. Taking this a step further, we also obtain integrality gaps for a semidefinite programming relaxation matching the integrality gaps of the earthmover linear program. Prior to this work, there was an intriguing possibility of obtaining better approximation factors for labeling problems via semidefinite programming.
Earth Mover’s Distance based Similarity Search at Scale
"... Earth Mover’s Distance (EMD), as a similarity measure, has received a lot of attention in the fields of multimedia and probabilistic databases, computer vision, image retrieval, machine learning, etc. EMD on multidimensional histograms provides better distinguishability between the objects approx ..."
Abstract
 Add to MetaCart
(Show Context)
Earth Mover’s Distance (EMD), as a similarity measure, has received a lot of attention in the fields of multimedia and probabilistic databases, computer vision, image retrieval, machine learning, etc. EMD on multidimensional histograms provides better distinguishability between the objects approximated by the histograms (e.g., images), compared to classic measures like Euclidean distance. Despite its usefulness, EMD has a high computational cost; therefore, a number of effective filtering methods have been proposed, to reduce the pairs of histograms for which the exact EMD has to be computed, during similarity search. Still, EMD calculations in the refinement step remain the bottleneck of the whole similarity search process. In this paper, we focus on optimizing the refinement phase of EMDbased similarity search by (i) adapting an efficient mincost flow algorithm (SIA) for EMD computation, (ii) proposing a dynamic distance bound, which can be used to terminate an EMD refinement early, and (iii) proposing a dynamic refinement order for the candidates which, paired with a concurrent EMD refinement strategy, reduces the amount of needless computations. Our proposed techniques are orthogonal to and can be easily integrated with the stateoftheart filtering techniques, reducing the cost of EMDbased similarity queries by orders of magnitude. 1.