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152
Aggregating inconsistent information: ranking and clustering
, 2005
"... We address optimization problems in which we are given contradictory pieces of input information and the goal is to find a globally consistent solution that minimizes the number of disagreements with the respective inputs. Specifically, the problems we address are rank aggregation, the feedback arc ..."
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Cited by 229 (17 self)
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We address optimization problems in which we are given contradictory pieces of input information and the goal is to find a globally consistent solution that minimizes the number of disagreements with the respective inputs. Specifically, the problems we address are rank aggregation, the feedback arc set problem on tournaments, and correlation and consensus clustering. We show that for all these problems (and various weighted versions of them), we can obtain improved approximation factors using essentially the same remarkably simple algorithm. Additionally, we almost settle a longstanding conjecture of BangJensen and Thomassen and show that unless NP⊆BPP, there is no polynomial time algorithm for the problem of minimum feedback arc set in tournaments.
Limits of dense graph sequences
 J. Combin. Theory Ser. B
"... We show that if a sequence of dense graphs Gn has the property that for every fixed graph F, the density of copies of F in Gn tends to a limit, then there is a natural “limit object”, namely a symmetric measurable function W: [0,1] 2 → [0, 1]. This limit object determines all the limits of subgraph ..."
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Cited by 208 (16 self)
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We show that if a sequence of dense graphs Gn has the property that for every fixed graph F, the density of copies of F in Gn tends to a limit, then there is a natural “limit object”, namely a symmetric measurable function W: [0,1] 2 → [0, 1]. This limit object determines all the limits of subgraph densities. Conversely, every such function arises as a limit object. Along the lines we introduce a rather general model of random graphs, which seems to be interesting on its own right. 1
Approximating the cutnorm via Grothendieck’s inequality
 Proc. of the 36 th ACM STOC
, 2004
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Efficient sampling algorithm for estimating subgraph concentrations and detecting network motifs
 Bioinformatics
, 2004
"... Biological and engineered networks have recently been shown to display network motifs: a small set of characteristic patterns which occur much more frequently than in randomized networks with the same degree sequence. Network motifs were demonstrated to play key information processing roles in biolo ..."
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Cited by 121 (0 self)
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Biological and engineered networks have recently been shown to display network motifs: a small set of characteristic patterns which occur much more frequently than in randomized networks with the same degree sequence. Network motifs were demonstrated to play key information processing roles in biological regulation networks. Existing algorithms for detecting network motifs act by exhaustively enumerating all subgraphs with a given number of nodes in the network. The runtime of such full enumeration algorithms increases strongly with network size. Here we present a novel algorithm that allows estimation of subgraph concentrations and detection of network motifs at a run time that is asymptotically independent of the network size. This algorithm is based on random sampling of subgraphs. Network motifs are detected with a surprisingly small number of samples in a wide variety of networks. Our method can be applied to estimate the concentrations of larger subgraphs in larger networks than was previously possible with full enumeration algorithms. We present results for highorder motifs in several biological networks and discuss their possible functions. Availability: A software tool for estimating subgraph concentrations and detecting network motifs (mfinder 2.0) and further information is available at:
The counting lemma for regular kuniform hypergraphs
, 2004
"... Szemerédi’s Regularity Lemma proved to be a powerful tool in the area of extremal graph theory. Many of its applications are based on its accompanying Counting Lemma: If G is an ℓpartite graph with V (G) = V1 ∪ · · · ∪ Vℓ and Vi  = n for all i ∈ [ℓ], and all pairs (Vi, Vj) are εregular of ..."
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Cited by 108 (14 self)
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Szemerédi’s Regularity Lemma proved to be a powerful tool in the area of extremal graph theory. Many of its applications are based on its accompanying Counting Lemma: If G is an ℓpartite graph with V (G) = V1 ∪ · · · ∪ Vℓ and Vi  = n for all i ∈ [ℓ], and all pairs (Vi, Vj) are εregular of density d for ℓ 1 ≤ i < j ≤ ℓ, then G contains (1 ± fℓ(ε))d
Testing that distributions are close
 In IEEE Symposium on Foundations of Computer Science
, 2000
"... Given two distributions over an n element set, we wish to check whether these distributions are statistically close by only sampling. We give a sublinear algorithm which uses O(n 2/3 ɛ −4 log n) independent samples from each distribution, runs in time linear in the sample size, makes no assumptions ..."
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Cited by 101 (18 self)
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Given two distributions over an n element set, we wish to check whether these distributions are statistically close by only sampling. We give a sublinear algorithm which uses O(n 2/3 ɛ −4 log n) independent samples from each distribution, runs in time linear in the sample size, makes no assumptions about the structure of the distributions, and distinguishes the cases ɛ when the distance between the distributions is small (less than max ( 2 32 3 √ n, ɛ 4 √)) or large (more n than ɛ) in L1distance. We also give an Ω(n 2/3 ɛ −2/3) lower bound. Our algorithm has applications to the problem of checking whether a given Markov process is rapidly mixing. We develop sublinear algorithms for this problem as well.
Subexponential algorithms for Unique Games and related problems
 IN 51 ST IEEE FOCS
, 2010
"... We give subexponential time approximation algorithms for the unique games and the small set expansion problems. Specifically, for some absolute constant c, we give: 1. An exp(kn ε)time algorithm that, given as input a kalphabet unique game on n variables that has an assignment satisfying 1 − ε c f ..."
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Cited by 82 (7 self)
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We give subexponential time approximation algorithms for the unique games and the small set expansion problems. Specifically, for some absolute constant c, we give: 1. An exp(kn ε)time algorithm that, given as input a kalphabet unique game on n variables that has an assignment satisfying 1 − ε c fraction of its constraints, outputs an assignment satisfying 1 − ε fraction of the constraints. 2. An exp(n ε /δ)time algorithm that, given as input an nvertex regular graph that has a set S of δn vertices with edge expansion at most ε c, outputs a set S ′ of at most δn vertices with edge expansion at most ε. We also obtain a subexponential algorithm with improved approximation for the MultiCut problem, as well as subexponential algorithms with improved approximations to MaxCut, SparsestCut and Vertex Cover on some interesting subclasses of instances. Khot’s Unique Games Conjecture (UGC) states that it is NPhard to achieve approximation guarantees such as ours for unique games. While our results stop short of refusing the UGC, they do suggest that Unique Games is significantly easier than NPhard problems such as 3SAT,3LIN, Label Cover and more, that are believed not to have a subexponential algorithm achieving a nontrivial approximation ratio. The main component in our algorithms is a new result on graph decomposition that may have other applications. Namely we show that for every δ> 0 and a regular nvertex graph G, by changing at most δ fraction of G’s edges, one can break G into disjoint parts so that the induced graph on each part has at most n ε eigenvalues larger than 1 − η (where ε, η depend polynomially on δ). Our results are based on combining this decomposition with previous algorithms for unique games on graphs with few large eigenvalues (Kolla and Tulsiani 2007, Kolla 2010).
Norm convergence of multiple ergodic averages for commuting transformations
, 2007
"... Let T1,..., Tl: X → X be commuting measurepreserving transformations on a probability space (X, X, µ). We show that the multiple ergodic averages 1 PN−1 N n=0 f1(T n 1 x)... fl(T n l x) are convergent in L2 (X, X, µ) as N → ∞ for all f1,..., fl ∈ L ∞ (X, X, µ); this was previously established fo ..."
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Cited by 81 (4 self)
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Let T1,..., Tl: X → X be commuting measurepreserving transformations on a probability space (X, X, µ). We show that the multiple ergodic averages 1 PN−1 N n=0 f1(T n 1 x)... fl(T n l x) are convergent in L2 (X, X, µ) as N → ∞ for all f1,..., fl ∈ L ∞ (X, X, µ); this was previously established for l = 2 by Conze and Lesigne [2] and for general l assuming some additional ergodicity hypotheses on the maps Ti and TiT −1 j by Frantzikinakis and Kra [3] (with the l = 3 case of this result established earlier in [29]). Our approach is combinatorial and finitary in nature, inspired by recent developments regarding the hypergraph regularity and removal lemmas, although we will not need the full strength of those lemmas. In particular, the l = 2 case of our arguments are a finitary analogue of those in [2].
Property Testing
 Handbook of Randomized Computing, Vol. II
, 2000
"... this technical aspect (as in the boundeddegree model the closest graph having the property must have at most dN edges and degree bound d as well). ..."
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Cited by 75 (10 self)
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this technical aspect (as in the boundeddegree model the closest graph having the property must have at most dN edges and degree bound d as well).
Szemerédi’s Lemma for the analyst
 J. GEOM. FUNC. ANAL
, 2006
"... Szemerédi’s Regularity Lemma is a fundamental tool in graph theory: it has many applications to extremal graph theory, graph property testing, combinatorial number theory, etc. The goal of this paper is to point out that Szemerédi’s Lemma can be thought of as a result in analysis. We show three diff ..."
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Cited by 69 (9 self)
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Szemerédi’s Regularity Lemma is a fundamental tool in graph theory: it has many applications to extremal graph theory, graph property testing, combinatorial number theory, etc. The goal of this paper is to point out that Szemerédi’s Lemma can be thought of as a result in analysis. We show three different analytic interpretations.