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14
Minmax graph partitioning and small set expansion
, 2011
"... We study graph partitioning problems from a minmax perspective, in which an input graph on n vertices should be partitioned into k parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main versions we consider are: (i) the k parts need to be of equal s ..."
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Cited by 14 (2 self)
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We study graph partitioning problems from a minmax perspective, in which an input graph on n vertices should be partitioned into k parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main versions we consider are: (i) the k parts need to be of equal size, and (ii) the parts must separate a set of k given terminals. We consider a common generalization of these two problems, and design for it an O ( √ log n log k)approximation algorithm. This improves over an O(log 2 n) approximation for the second version due to Svitkina and Tardos [22], and roughly O(k log n) approximation for the first version that follows from other previous work. We also give an improved O(1)approximation algorithm for graphs that exclude any fixed minor. Our algorithm uses a new procedure for solving the SmallSet Expansion problem. In this problem, we are given a graph G and the goal is to find a nonempty set S ⊆ V of size S  ≤ ρn with minimum edgeexpansion. We give an O ( √ log n log (1/ρ)) bicriteria approximation algorithm for the general case of SmallSet Expansion, and O(1) approximation algorithm for graphs that exclude any fixed minor.
Approximating Sparsest Cut in Graphs of Bounded Treewidth
"... We give the first constantfactor approximation algorithm for SparsestCut with general demands in bounded treewidth graphs. In contrast to previous algorithms, which rely on the flowcut gap and/or metric embeddings, our approach exploits the SheraliAdams hierarchy of linear programming relaxation ..."
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Cited by 5 (2 self)
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We give the first constantfactor approximation algorithm for SparsestCut with general demands in bounded treewidth graphs. In contrast to previous algorithms, which rely on the flowcut gap and/or metric embeddings, our approach exploits the SheraliAdams hierarchy of linear programming relaxations.
Sparsest cut on bounded treewidth graphs: Algorithms and hardness results
 In 45th Annual ACM Symposium on Symposium on Theory of Computing
, 2013
"... We give a 2approximation algorithm for NonUniform Sparsest Cut that runs in time nO(k), where k is the treewidth of the graph. This improves on the previous 22 kapproximation in time poly(n)2O(k) due to Chlamtác ̌ et al. [CKR10]. To complement this algorithm, we show the following hardness resul ..."
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Cited by 4 (0 self)
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We give a 2approximation algorithm for NonUniform Sparsest Cut that runs in time nO(k), where k is the treewidth of the graph. This improves on the previous 22 kapproximation in time poly(n)2O(k) due to Chlamtác ̌ et al. [CKR10]. To complement this algorithm, we show the following hardness results: If the NonUniform Sparsest Cut problem has a ρapproximation for seriesparallel graphs (where ρ ≥ 1), then the MaxCut problem has an algorithm with approximation factor arbitrarily close to 1/ρ. Hence, even for such restricted graphs (which have treewidth 2), the Sparsest Cut problem is NPhard to approximate better than 17/16 − ε for ε> 0; assuming the Unique Games Conjecture the hardness becomes 1/αGW − ε. For graphs with large (but constant) treewidth, we show a hardness result of 2 − ε assuming the Unique Games Conjecture. Our algorithm rounds a linear program based on (a subset of) the SheraliAdams lift of the standard Sparsest Cut LP. We show that even for treewidth2 graphs, the LP has an integrality gap close to 2 even after polynomially many rounds of SheraliAdams. Hence our approach cannot be improved even on such restricted graphs without using a stronger relaxation. 1
Low Dimensional Embeddings of Doubling Metrics
, 2013
"... We study several embeddings of doubling metrics into low dimensional normed spaces, in particular into `2 and `∞. Doubling metrics are a robust class of metric spaces that have low intrinsic dimension, and often occur in applications. Understanding the dimension required for a concise representation ..."
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Cited by 2 (2 self)
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We study several embeddings of doubling metrics into low dimensional normed spaces, in particular into `2 and `∞. Doubling metrics are a robust class of metric spaces that have low intrinsic dimension, and often occur in applications. Understanding the dimension required for a concise representation of such metrics is a fundamental open problem in the area of metric embedding. Here we show that the nvertex Laakso graph can be embedded into constant dimensional `2 with the best possible distortion, which has implications for possible approaches to the above problem. Since arbitrary doubling metrics require high distortion for embedding into `2 and even into `1, we turn to the ` ∞ space that enables us to obtain arbitrarily small distortion. We show embeddings of doubling metrics and their ”snowflakes ” into low dimensional ` ∞ space that simplify and extend previous results. 1
MINMAX GRAPH PARTITIONING AND SMALL SET EXPANSION∗
"... Abstract. We study graph partitioning problems from a minmax perspective, in which an input graph on n vertices should be partitioned into k parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main versions we consider are where the k parts need to be ..."
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Abstract. We study graph partitioning problems from a minmax perspective, in which an input graph on n vertices should be partitioned into k parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main versions we consider are where the k parts need to be of equal size, and where they must separate a set of k given terminals. We consider a common generalization of these two problems, and design for it an O( logn log k) approximation algorithm. This improves over an O(log2 n) approximation for the second version due to Svitkina and Tardos [Minmax multiway cut, in APPROXRANDOM, 2004, Springer, Berlin, 2004], and roughly O(k logn) approximation for the first version that follows from other previous work. We also give an O(1) approximation algorithm for graphs that exclude any fixed minor. Our algorithm uses a new procedure for solving the smallset expansion problem. In this problem, we are given a graph G and the goal is to find a nonempty set S ⊆ V of size S  ≤ ρn with minimum edge expansion. We give an O( logn log (1/ρ)) bicriteria approximation algorithm for smallset expansion in general graphs, and an improved factor of O(1) for graphs that exclude any fixed minor.