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190
Property Testing and its connection to Learning and Approximation
"... We study the question of determining whether an unknown function has a particular property or is fflfar from any function with that property. A property testing algorithm is given a sample of the value of the function on instances drawn according to some distribution, and possibly may query the fun ..."
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Cited by 498 (68 self)
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We study the question of determining whether an unknown function has a particular property or is fflfar from any function with that property. A property testing algorithm is given a sample of the value of the function on instances drawn according to some distribution, and possibly may query the function on instances of its choice. First, we establish some connections between property testing and problems in learning theory. Next, we focus on testing graph properties, and devise algorithms to test whether a graph has properties such as being kcolorable or having a aeclique (clique of density ae w.r.t the vertex set). Our graph property testing algorithms are probabilistic and make assertions which are correct with high probability, utilizing only poly(1=ffl) edgequeries into the graph, where ffl is the distance parameter. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph which corre...
Multicommodity maxflow mincut theorems and their use in designing approximation algorithms
 J. ACM
, 1999
"... Abstract. In this paper, we establish maxflow mincut theorems for several important classes of multicommodity flow problems. In particular, we show that for any nnode multicommodity flow problem with uniform demands, the maxflow for the problem is within an O(log n) factor of the upper bound imp ..."
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Cited by 370 (6 self)
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Abstract. In this paper, we establish maxflow mincut theorems for several important classes of multicommodity flow problems. In particular, we show that for any nnode multicommodity flow problem with uniform demands, the maxflow for the problem is within an O(log n) factor of the upper bound implied by the mincut. The result (which is existentially optimal) establishes an important analogue of the famous 1commodity maxflow mincut theorem for problems with multiple commodities. The result also has substantial applications to the field of approximation algorithms. For example, we use the flow result to design the first polynomialtime (polylog ntimesoptimal) approximation algorithms for wellknown NPhard optimization problems such as graph partitioning, mincut linear arrangement, crossing number, VLSI layout, and minimum feedback arc set. Applications of the flow results to path routing problems, network reconfiguration, communication in distributed networks, scientific computing and rapidly mixing Markov chains are also described in the paper.
Correlation Clustering
 MACHINE LEARNING
, 2002
"... We consider the following clustering problem: we have a complete graph on # vertices (items), where each edge ### ## is labeled either # or depending on whether # and # have been deemed to be similar or different. The goal is to produce a partition of the vertices (a clustering) that agrees as mu ..."
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Cited by 329 (4 self)
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We consider the following clustering problem: we have a complete graph on # vertices (items), where each edge ### ## is labeled either # or depending on whether # and # have been deemed to be similar or different. The goal is to produce a partition of the vertices (a clustering) that agrees as much as possible with the edge labels. That is, we want a clustering that maximizes the number of # edges within clusters, plus the number of edges between clusters (equivalently, minimizes the number of disagreements: the number of edges inside clusters plus the number of # edges between clusters). This formulation is motivated from a document clustering problem in which one has a pairwise similarity function # learned from past data, and the goal is to partition the current set of documents in a way that correlates with # as much as possible; it can also be viewed as a kind of "agnostic learning" problem. An interesting
The Dense kSubgraph Problem
 Algorithmica
, 1999
"... This paper considers the problem of computing the dense kvertex subgraph of a given graph, namely, the subgraph with the most edges. An approximation algorithm is developed for the problem, with approximation ratio O(n ffi ), for some ffi ! 1=3. 1 Introduction We study the dense ksubgraph (D ..."
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Cited by 205 (12 self)
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This paper considers the problem of computing the dense kvertex subgraph of a given graph, namely, the subgraph with the most edges. An approximation algorithm is developed for the problem, with approximation ratio O(n ffi ), for some ffi ! 1=3. 1 Introduction We study the dense ksubgraph (DkS) maximization problem, of computing the dense k vertex subgraph of a given graph. That is, on input a graph G and a parameter k, we are interested in finding a set of k vertices with maximum average degree in the subgraph induced by this set. As this problem is NPhard (say, by reduction from Clique), we consider approximation algorithms for this problem. We obtain a polynomial time algorithm that on any input (G; k) returns a subgraph of size k whose average degree is within a factor of at most n ffi from the optimum solution, where n is the number of vertices in the input graph G, and ffi ! 1=3 is some universal constant. Unfortunately, we are unable to present a complementary negati...
Quick Approximation to Matrices and Applications
, 1998
"... We give algorithms to find the following simply described approximation to a given matrix. Given an m \Theta n matrix A with entries between say1 and 1, and an error parameter ffl between 0 and 1, we find a matrix D (implicitly) which is the sum of O(1=ffl 2 ) simple rank 1 matrices so that the ..."
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Cited by 151 (7 self)
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We give algorithms to find the following simply described approximation to a given matrix. Given an m \Theta n matrix A with entries between say1 and 1, and an error parameter ffl between 0 and 1, we find a matrix D (implicitly) which is the sum of O(1=ffl 2 ) simple rank 1 matrices so that the sum of entries of any submatrix (among the 2 m+n ) of (A \Gamma D) is at most fflmn in absolute value. Our algorithm takes time dependent only on ffl and the allowed probability of failure (not on m;n). We draw on two lines of research to develop the algorithms: one is built around the fundamental Regularity Lemma of Szemerédi in Graph Theory and the constructive version of Alon, Duke, Leffman, Rödl and Yuster. The second one is from the papers of Arora, Karger and Karpinski, Fernandez de la Vega and most directly Goldwasser, Goldreich and Ron who develop approximation algorithms for a set of graph problems, typical of which is the maximum cut problem. ?From our matrix approximation, the...
A Polylogarithmic Approximation of the Minimum Bisection
, 2001
"... A bisection of a graph with n vertices is a partition of its vertices into two sets, each of size n=2. The bisection cost is the number of edges connecting the two sets. ..."
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Cited by 89 (7 self)
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A bisection of a graph with n vertices is a partition of its vertices into two sets, each of size n=2. The bisection cost is the number of edges connecting the two sets.
Segmentation problems
, 2004
"... We study a novel genre of optimization problems, which we call segmentation problems, motivated in part by certain aspects of clustering and data mining. For any classical optimization problem, the corresponding segmentation problem seeks to partition a set of cost vectors into several segments, s ..."
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Cited by 86 (5 self)
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We study a novel genre of optimization problems, which we call segmentation problems, motivated in part by certain aspects of clustering and data mining. For any classical optimization problem, the corresponding segmentation problem seeks to partition a set of cost vectors into several segments, so that the overall cost is optimized. We focus on two natural and interesting (but MAXSNPcomplete) problems in this class, the HYPERCUBE SEGMENTATION PROBLEM and the CATALOG SEGMENTATION PROBLEM, and present approximation algorithms for them. We also present a general greedy scheme, which can be specialized to approximate any segmentation problem.
Algorithms for Graph Partitioning on the Planted Partition Model
, 1999
"... The NPhard graph bisection problem is to partition the nodes of an undirected graph into two equalsized groups so as to minimize the number of edges that cross the partition. The more general graph lpartition problem is to partition the nodes of an undirected graph into l equalsized groups so as ..."
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Cited by 86 (0 self)
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The NPhard graph bisection problem is to partition the nodes of an undirected graph into two equalsized groups so as to minimize the number of edges that cross the partition. The more general graph lpartition problem is to partition the nodes of an undirected graph into l equalsized groups so as to minimize the total number of edges that cross between groups. We present a simple, lineartime algorithm for the graph lpartition problem and analyze it on a random "planted lpartition" model. In this model, the n nodes of a graph are partitioned into l groups, each of size n=l; two nodes in the same group are connected by an edge with some probability p, and two nodes in different groups are connected by an edge with some probability r ! p. We show that if p \Gamma r n \Gamma1=2+ffl for some constant ffl, then the algorithm finds the optimal partition with probability 1 \Gamma exp(\Gamman \Theta(ffl) ). 1