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16
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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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
Energy Minimization Under Constraints on Label Counts
"... Many computer vision problems such as object segmentation or reconstruction can be formulated in terms of labeling a set of pixels or voxels. In certain scenarios, we may know the number of pixels or voxels which can be assigned to a particular label. For instance, in the reconstruction problem, w ..."
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Cited by 11 (3 self)
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Many computer vision problems such as object segmentation or reconstruction can be formulated in terms of labeling a set of pixels or voxels. In certain scenarios, we may know the number of pixels or voxels which can be assigned to a particular label. For instance, in the reconstruction problem, we may know size of the object to be reconstructed. Such label count constraints are extremely powerful and have recently been shown to result in good solutions for many vision problems. Traditional energy minimization algorithms used in vision cannot handle label count constraints. This paper proposes a novel algorithm for minimizing energy functions under constraints on the number of variables which can be assigned to a particular label. Our algorithm is deterministic in nature and outputs εapproximate solutions for all possible counts of labels. We also develop a variant of the above algorithm which is much faster, produces solutions under almost all label count constraints, and can be applied to all submodular quadratic pseudoboolean functions. We evaluate the algorithm on the twolabel (foreground/background) image segmentation problem and compare its performance with the stateoftheart parametric maximum flow and maxsum diffusion based algorithms. Experimental results show that our method is practical and is able to generate impressive segmentation results in reasonable time.
On Earthmover Distance, Metric Labeling, and 0Extension
, 2006
"... We study the fundamental classification problems 0Extension and Metric Labeling. 0Extension is closely ..."
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Cited by 9 (0 self)
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We study the fundamental classification problems 0Extension and Metric Labeling. 0Extension is closely
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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Cited by 7 (0 self)
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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 directed routing with congestion
 Electronic Colloquium on Computational Complexity
, 2006
"... Given a graph G and a collection of sourcesink pairs in G, what is the least integer c such that each source can be connected by a path to its sink, with at most c paths going through an edge? This is known as the congestion minimization problem, and the quantity c is called the congestion. Congest ..."
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Given a graph G and a collection of sourcesink pairs in G, what is the least integer c such that each source can be connected by a path to its sink, with at most c paths going through an edge? This is known as the congestion minimization problem, and the quantity c is called the congestion. Congestion minimization is one of the most wellstudied NPhard optimization problems. It is wellknown that the elegant randomized rounding technique of Raghavan and Thompson can be used to obtain a solution with congestion at most c ∗ log n + O( log log n) where c ∗ is the optimal congestion. In this paper we show that there exists a δ> 0 such that no polynomialtime algorithm can guarantee a solution with congestion c ∗ δ log n + ( ) unless NP is log log n contained in ZPTIME(nlog log n). We also study the directed edgedisjoint paths (EDP) problem with congestion. The input
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
Capacitated Metric Labeling
, 2011
"... We introduce Capacitated Metric Labeling. As in Metric Labeling, we are given a weighted graph G = (V, E), a label set L, a semimetric dL on this label set, and an assignment cost function φ: V × L → ℜ +. The goal in Metric Labeling is to find an assignment f: V → L that minimizes a particular twoc ..."
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We introduce Capacitated Metric Labeling. As in Metric Labeling, we are given a weighted graph G = (V, E), a label set L, a semimetric dL on this label set, and an assignment cost function φ: V × L → ℜ +. The goal in Metric Labeling is to find an assignment f: V → L that minimizes a particular twocost function. Here we add the additional restriction that each label ti receive at most li nodes, and we refer to this problem as Capacitated Metric Labeling. Allowing the problem to specify capacities on each label allows the problem to more faithfully represent the classification problems that Metric Labeling is intended to model. Our main positive result is a polynomialtime, O(log V )approximation algorithm when the number of labels is fixed, which is the most natural parameter range for classification problems. We also prove that it is impossible to approximate the value of an instance of Capacitated Metric Labeling to within any finite factor, if P ̸ = NP. Yet this does not address the more interesting question of how hard Capacitated Metric Labeling is to approximate when we are allowed to violate capacities. To study this question, we introduce the notion of the “congestion” of an instance of Capacitated Metric Labeling. We prove that (under certain complexity assumptions) there is no polynomialtime approximation algorithm that can approximate the congestion to within O((log L) 1/2−ɛ) (for any ɛ> 0) and this implies as a corollary that any polynomialtime approximation algorithm that achieves a finite approximation ratio must multiplicatively violate the label capacities by Ω((log L) 1/2−ɛ). We also give a O(log L)approximation algorithm for congestion.
Parallel Imaging Problem
"... Metric Labeling problems have been introduced as a model for understanding noisy data with pairwise relations between the data points. One application of labeling problems with pairwise relations is image understanding, where the underlying assumption is that physically close pixels are likely to ..."
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Metric Labeling problems have been introduced as a model for understanding noisy data with pairwise relations between the data points. One application of labeling problems with pairwise relations is image understanding, where the underlying assumption is that physically close pixels are likely to belong to the same object. In this paper we consider a variant of this problem, we will call Parallel Imaging, where instead of directly observing the noisy data, the data undergoes a simple linear transformation first, such as adding different images. This class of problems arises in a wide range of imaging problems. Our study has been motivated by the Parallel Imaging problem in Magnetic Resonance Image (MRI) reconstruction. We give a constant factor approximation algorithm for the case of speedup of two with the truncated linear metric, motivated by the MRI reconstruction problem. Our method uses local search and graph cut techniques.
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.