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99
Improved Combinatorial Algorithms for the Facility Location and kMedian Problems
 In Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science
, 1999
"... We present improved combinatorial approximation algorithms for the uncapacitated facility location and kmedian problems. Two central ideas in most of our results are cost scaling and greedy improvement. We present a simple greedy local search algorithm which achieves an approximation ratio of 2:414 ..."
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Cited by 227 (11 self)
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We present improved combinatorial approximation algorithms for the uncapacitated facility location and kmedian problems. Two central ideas in most of our results are cost scaling and greedy improvement. We present a simple greedy local search algorithm which achieves an approximation ratio of 2:414 + in ~ O(n 2 =) time. This also yields a bicriteria approximation tradeoff of (1 +; 1+ 2=) for facility cost versus service cost which is better than previously known tradeoffs and close to the best possible. Combining greedy improvement and cost scaling with a recent primal dual algorithm for facility location due to Jain and Vazirani, we get an approximation ratio of 1.853 in ~ O(n 3 ) time. This is already very close to the approximation guarantee of the best known algorithm which is LPbased. Further, combined with the best known LPbased algorithm for facility location, we get a very slight improvement in the approximation factor for facility location, achieving 1.728....
Primaldual algorithms for connected facility location problems
 Algorithmica
, 2002
"... We consider the Connected Facility Location problem. We are given a graph G = (V, E) with costs {ce} on the edges, a set of facilities F ⊆ V, and a set of clients D ⊆ V. Facility i has a facility opening cost fi and client j has dj units of demand. We are also given a parameter M ≥ 1. A solution ope ..."
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Cited by 79 (9 self)
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We consider the Connected Facility Location problem. We are given a graph G = (V, E) with costs {ce} on the edges, a set of facilities F ⊆ V, and a set of clients D ⊆ V. Facility i has a facility opening cost fi and client j has dj units of demand. We are also given a parameter M ≥ 1. A solution opens some facilities, say F, assigns each client j to an open facility i(j), and connects the open facilities by a Steiner tree T. The total cost incurred is � i∈F fi + � j∈D djci(j)j + M � e∈T ce. We want a solution of minimum cost. A special case of this problem is when all opening costs are 0 and facilities may be opened anywhere, i.e., F = V. If we know a facility v that is open, then the problem becomes a special case of the singlesink buyatbulk problem with two cable types, also known as the rentorbuy problem. We give the first primaldual algorithms for these problems and achieve the best known approximation guarantees. We give a 8.55approximation algorithm for the connected facility location problem and a 4.55approximation algorithm for the rentorbuy problem. Previously the best approximation factors for these problems were 10.66 and 9.001 respectively [8]. Further, these results were not combinatorial — they were obtained by solving an exponential size linear programming relaxation. Our algorithm integrates the primaldual approaches for the facility location problem [11] and the Steiner tree problem [1, 3]. We also consider the connected kmedian problem and give a constantfactor approximation by using our primaldual algorithm for connected facility location. We generalize our results to an edge capacitated variant of these problems and give a constantfactor approximation for these variants.
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...
Approximate labeling via graphcuts based on linear programming
 In Pattern Analysis and Machine Intelligence
, 2007
"... A new framework is presented for both understanding and developing graphcut based combinatorial algorithms suitable for the approximate optimization of a very wide class of MRFs that are frequently encountered in computer vision. The proposed framework utilizes tools from the duality theory of line ..."
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Cited by 74 (8 self)
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A new framework is presented for both understanding and developing graphcut based combinatorial algorithms suitable for the approximate optimization of a very wide class of MRFs that are frequently encountered in computer vision. The proposed framework utilizes tools from the duality theory of linear programming in order to provide an alternative and more general view of stateoftheart techniques like the αexpansion algorithm, which is included merely as a special case. Moreover, contrary to αexpansion, the derived algorithms generate solutions with guaranteed optimality properties for a much wider class of problems, e.g. even for MRFs with nonmetric potentials. In addition, they are capable of providing perinstance suboptimality bounds in all occasions, including discrete Markov Random Fields with an arbitrary potential function. These bounds prove to be very tight in practice (i.e. very close to 1), which means that the resulting solutions are almost optimal. Our algorithms ’ effectiveness is demonstrated by presenting experimental results on a variety of low level vision tasks, such as stereo matching, image restoration, image completion and optical flow estimation, as well as on synthetic problems.
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.
Hardness of buyatbulk network design
 In Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
, 2004
"... ..."
Approximation algorithms for nonuniform buyatbulk network design problems
 Proc. of IEEE FOCS
"... Abstract. Buyatbulk network design problems arise in settings where the costs for purchasing or installing equipment exhibit economies of scale. The objective is to build a network of cheapest cost to support a given multicommodity flow demand between node pairs. We present approximation algorith ..."
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Cited by 56 (13 self)
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Abstract. Buyatbulk network design problems arise in settings where the costs for purchasing or installing equipment exhibit economies of scale. The objective is to build a network of cheapest cost to support a given multicommodity flow demand between node pairs. We present approximation algorithms for buyatbulk network design problems with costs on both edges and nodes of an undirected graph. Our main result is the first polylogarithmic approximation ratio for the nonuniform problem that allows different cost functions on each edge and node; the ratio we achieve is O(log4 h) where h is the number of demand pairs. In addition we present an O(log h) approximation for the single sink problem. Polylogarithmic ratios for some related problems are also obtained. Our algorithm for the multicommodity problem is obtained via a reduction to the single source problem using the notion of junction trees. We believe that this presents a simple yet useful general technique for network design problems. Key words. Nonuniform buyatbulk, network design, approximation algorithm, concave cost, network flow, economies of scale AMS subject classifications. 68Q25, 68W25, 90C27, 90C59 1. Introduction. Network
An efficient algorithm for image segmentation, Markov random fields and related problems
 Journal of the ACM
, 2001
"... Abstract. Problems of statistical inference involve the adjustment of sample observations so they fit some a priori rank requirements, or order constraints. In such problems, the objective is to minimize the deviation cost function that depends on the distance between the observed value and the modi ..."
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Cited by 52 (13 self)
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Abstract. Problems of statistical inference involve the adjustment of sample observations so they fit some a priori rank requirements, or order constraints. In such problems, the objective is to minimize the deviation cost function that depends on the distance between the observed value and the modify value. In Markov random field problems, there is also a pairwise relationship between the objects. The objective in Markov random field problem is to minimize the sum of the deviation cost function and a penalty function that grows with the distance between the values of related pairs— separation function. We discuss Markov random fields problems in the context of a representative application—the image segmentation problem. In this problem, the goal is to modify color shades assigned to pixels of an image so that the penalty function consisting of one term due to the deviation from the initial color shade and a second term that penalizes differences in assigned values to neighboring pixels is minimized. We present here an algorithm that solves the problem in polynomial time when the deviation function is convex and separation function is linear; and in strongly polynomial time when the deviation cost function is linear, quadratic or piecewise linear convex with few pieces (where “few” means a number exponential in a polynomial function of the number of variables and constraints).