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Embeddings of negativetype metrics and an improved approximation to generalized sparsest cut
 IN SODA ‘05: PROCEEDINGS OF THE SIXTEENTH ANNUAL ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS
, 2005
"... In this paper, we study the metrics of negative type, which are metrics (V,d) such that √ d is an Euclidean metric; these metrics are thus also known as “ℓ2squared” metrics. We show how to embed npoint negativetype metrics into Euclidean space ℓ2 with distortion D = O(log 3/4 n). This embedding re ..."
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Cited by 44 (0 self)
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In this paper, we study the metrics of negative type, which are metrics (V,d) such that √ d is an Euclidean metric; these metrics are thus also known as “ℓ2squared” metrics. We show how to embed npoint negativetype metrics into Euclidean space ℓ2 with distortion D = O(log 3/4 n). This embedding result, in turn, implies an O(log 3/4 k)approximation algorithm for the Sparsest Cut problem with nonuniform demands. Another corollary we obtain is that npoint subsets of ℓ1 embed into ℓ2 with distortion O(log 3/4 n).
Applications of Cut Polyhedra
, 1992
"... We group in this paper, within a unified framework, many applications of the following polyhedra: cut, boolean quadric, hypermetric and metric polyhedra. We treat, in particular, the following applications: ffl ` 1  and L 1 metrics in functional analysis, ffl the maxcut problem, the Boole probl ..."
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Cited by 25 (2 self)
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We group in this paper, within a unified framework, many applications of the following polyhedra: cut, boolean quadric, hypermetric and metric polyhedra. We treat, in particular, the following applications: ffl ` 1  and L 1 metrics in functional analysis, ffl the maxcut problem, the Boole problem and multicommodity flow problems in combinatorial optimization, ffl lattice holes in geometry of numbers, ffl density matrices of manyfermions systems in quantum mechanics. We present some other applications, in probability theory, statistical data analysis and design theory.
A T_Xapproach to some results on cuts and metrics
 Advances in Applied Mathematics 19
, 1997
"... We give simple algorithmic proofs of some theorems of Papernov (1976) and Karzanov (1985,1990) on the packing of metrics by cuts. 1. ..."
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Cited by 6 (0 self)
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We give simple algorithmic proofs of some theorems of Papernov (1976) and Karzanov (1985,1990) on the packing of metrics by cuts. 1.
Recognition of the l 1 graphs with Complexity O(nm), or Football in a Hypercube
, 1995
"... We fill in the details of the algorithm sketched in [Sh] and determine its complexity. As a part of this main algorithm, we also describe an algorithm which recognizes graphs which are isometric subgraphs of halved cubes. We discuss possible further applications of the same ideas and give a nice exa ..."
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Cited by 5 (5 self)
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We fill in the details of the algorithm sketched in [Sh] and determine its complexity. As a part of this main algorithm, we also describe an algorithm which recognizes graphs which are isometric subgraphs of halved cubes. We discuss possible further applications of the same ideas and give a nice example of a non ` 1 graph allowing a highly isometric embedding into a halved cube. 1 Introduction For a set \Omega\Gamma let 2\Omega denote the set of all the subsets of \Omega\Gamma We turn 2\Omega into an ndimensional cube graph Q k , where k = j\Omega j, by making two subsets A and B adjacent whenever the symmetric difference A4B has size 1. The graph Q k is bipartite, the bipartite half of Q k being known as the halved cube graph (we denote it HQ k ). Therefore, HQ k can be defined as the graph on the even size subsets in 2\Omega , in which two such subsets A and B are adjacent whenever jA4Bj = 2. We usually identify a graph \Gamma with its set of vertices, and we use the same notati...
Fullerenes and Coordination Polyhedra versus HalfCubes Embeddings
, 1997
"... A fullerene F n is a 3regular (or cubic) polyhedral carbon molecule for which the n vertices  the carbons atoms  are arranged in 12 pentagons and ( n 2 \Gamma 10) hexagons. Only a finite number of fullerenes are expected to be, up to scale, isometrically embeddable into a hypercube. Looking fo ..."
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Cited by 2 (0 self)
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A fullerene F n is a 3regular (or cubic) polyhedral carbon molecule for which the n vertices  the carbons atoms  are arranged in 12 pentagons and ( n 2 \Gamma 10) hexagons. Only a finite number of fullerenes are expected to be, up to scale, isometrically embeddable into a hypercube. Looking for the list of such fullerenes, we first check the embeddability of all fullerenes F n for n ! 60 and of all preferable fullerenes C n for n ! 86 and their duals. Then, we consider some infinite families, including fullerenes with icosahedral symmetry, which describe virus capsids, onionlike metallic clusters and geodesic domes. Quasiembeddings and fullerene analogues are considered. We also present some results on chemically relevant polyhedra such as coordination polyhedra and cluster polyhedra. Finally we conjecture that the list of known embeddable fullerenes is complete and present its relevance to the Katsura model for vesicles cells. Contents 1 Introduction and Basic Properties 2 1...
Hypercube Embeddings and Designs
, 1994
"... This is a survey on hypercube embeddable semimetrics and the link with designs. We investigate, in particular, the variety of hypercube embeddings of the equidistant metric. For some parameters, it is linked with the question of existence of projective planes or Hadamard matrices. The problem of tes ..."
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This is a survey on hypercube embeddable semimetrics and the link with designs. We investigate, in particular, the variety of hypercube embeddings of the equidistant metric. For some parameters, it is linked with the question of existence of projective planes or Hadamard matrices. The problem of testing whether a semimetric is hypercube embeddable is NPhard in general. Several classes of semimetrics are described for which this problem can be solved in polynomial time. We also consider questions related to some necessary conditions for hypercube embeddability.
Notes taken by Periklis Papakonstantinou revised by Hamed Hatami
"... Lecture 7: Lower bounds on the embeddability in via expander graphs and some algorithmic connections to ..."
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Lecture 7: Lower bounds on the embeddability in via expander graphs and some algorithmic connections to
TXApproaches to Multiflows and Metrics By
"... This paper is an exposition of a unified approach to multiflow problems using certain polyhedral objects called tight spans or TXspaces. The tight span was introduced by Isbell and Dress, independent on the multiflow research. In the middle of 90’s, Karzanov and Chepoi explored the significance of ..."
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This paper is an exposition of a unified approach to multiflow problems using certain polyhedral objects called tight spans or TXspaces. The tight span was introduced by Isbell and Dress, independent on the multiflow research. In the middle of 90’s, Karzanov and Chepoi explored the significance of tight spans in the multiflow theory. We explain how the tight span derives minmax relations to multiflow problems and how its geometry affects discreteness issues of flows and potentials.
Metric packing for K3 + K3
, 2009
"... In this paper, we consider the metric packing problem for the commodity graph of disjoint two triangles K3 + K3, which is dual to the multiflow feasibility problem for the commodity graph K3 + K3. We prove a strengthening of Karzanov’s conjecture concerning quarterintegral packings by certain bipar ..."
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In this paper, we consider the metric packing problem for the commodity graph of disjoint two triangles K3 + K3, which is dual to the multiflow feasibility problem for the commodity graph K3 + K3. We prove a strengthening of Karzanov’s conjecture concerning quarterintegral packings by certain bipartite metrics.