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23
The geometry of graphs and some of its algorithmic applications
 COMBINATORICA
, 1995
"... In this paper we explore some implications of viewing graphs as geometric objects. This approach offers a new perspective on a number of graphtheoretic and algorithmic problems. There are several ways to model graphs geometrically and our main concern here is with geometric representations that res ..."
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Cited by 524 (19 self)
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In this paper we explore some implications of viewing graphs as geometric objects. This approach offers a new perspective on a number of graphtheoretic and algorithmic problems. There are several ways to model graphs geometrically and our main concern here is with geometric representations that respect the metric of the (possibly weighted) graph. Given a graph G we map its vertices to a normed space in an attempt to (i) Keep down the dimension of the host space and (ii) Guarantee a small distortion, i.e., make sure that distances between vertices in G closely match the distances between their geometric images. In this paper we develop efficient algorithms for embedding graphs lowdimensionally with a small distortion. Further algorithmic applications include: 0 A simple, unified approach to a number of problems on multicommodity flows, including the LeightonRae Theorem [29] and some of its extensions. 0 For graphs embeddable in lowdimensional spaces with a small distortion, we can find lowdiameter decompositions (in the sense of [4] and [34]). The parameters of the decomposition depend only on the dimension and the distortion and not on the size of the graph. 0 In graphs embedded this way, small balanced separators can be found efficiently. Faithful lowdimensional representations of statistical data allow for meaningful and efficient clustering, which is one of the most basic tasks in patternrecognition. For the (mostly heuristic) methods used
An O(log k) approximate mincut maxflow theorem and approximation algorithm
 SIAM J. COMPUT
, 1998
"... It is shown that the minimum cut ratio is within a factor of O(log k) of the maximum concurrent flow for kcommodity flow instances with arbitrary capacities and demands. This improves upon the previously bestknown bound of O(log 2 k) and is existentially tight, up to a constant factor. An algori ..."
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Cited by 129 (6 self)
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It is shown that the minimum cut ratio is within a factor of O(log k) of the maximum concurrent flow for kcommodity flow instances with arbitrary capacities and demands. This improves upon the previously bestknown bound of O(log 2 k) and is existentially tight, up to a constant factor. An algorithm for finding a cut with ratio within a factor of O(log k) of the maximum concurrent flow, and thus of the optimal mincut ratio, is presented.
Solving Quadratic (0,1)Problems by Semidefinite Programs and Cutting Planes
, 1996
"... We present computational experiments for solving quadratic (0, 1) problems. Our approach combines a semidefinite relaxation with a cutting plane technique, and is applied in a Branch and Bound setting. Our experiments indicate that this type of approach is very robust, and allows to solve many moder ..."
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Cited by 57 (7 self)
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We present computational experiments for solving quadratic (0, 1) problems. Our approach combines a semidefinite relaxation with a cutting plane technique, and is applied in a Branch and Bound setting. Our experiments indicate that this type of approach is very robust, and allows to solve many moderately sized problems, having say, less than 100 binary variables, in a routine manner.
Clustering with the connectivity kernel
 In NIPS
, 2004
"... Clustering aims at extracting hidden structure in dataset. While the problem of finding compact clusters has been widely studied in the literature, extracting arbitrarily formed elongated structures is considered a much harder problem. In this paper we present a novel clustering algorithm which tack ..."
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Cited by 34 (1 self)
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Clustering aims at extracting hidden structure in dataset. While the problem of finding compact clusters has been widely studied in the literature, extracting arbitrarily formed elongated structures is considered a much harder problem. In this paper we present a novel clustering algorithm which tackles the problem by a two step procedure: first the data are transformed in such a way that elongated structures become compact ones. In a second step, these new objects are clustered by optimizing a compactnessbased criterion. The advantages of the method over related approaches are threefold: (i) robustness properties of compactnessbased criteria naturally transfer to the problem of extracting elongated structures, leading to a model which is highly robust against outlier objects; (ii) the transformed distances induce a Mercer kernel which allows us to formulate a polynomial approximation scheme to the generally N Phard clustering problem; (iii) the new method does not contain free kernel parameters in contrast to methods like spectral clustering or meanshift clustering. 1
Clin d'Oeil on L_1Embeddable Planar Graphs
, 1996
"... In this note we present some properties of L1embeddable planar garphs. We show that every such graph G has a scale 2 embedding into a hypercube. Further, under some additional conditions we show that for a simple circuit C of G the subgraph H of G bounded by C is also L1embeddable. In many importa ..."
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Cited by 19 (2 self)
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In this note we present some properties of L1embeddable planar garphs. We show that every such graph G has a scale 2 embedding into a hypercube. Further, under some additional conditions we show that for a simple circuit C of G the subgraph H of G bounded by C is also L1embeddable. In many important cases, the length of C is the dimension of the smallest cube in which H has a scale embedding. Using these facts we establish the L1embeddability of a list of planar graphs.
On skeletons, diameters and volumes of metric polyhedra
 COMBINATORICS AND COMPUTER SCIENCE, LECTURE
"... We survey and present new geometric and combinatorial propertiez of some polyhedra with application in combinatorial optimization, for example, the maxcut and multicommodity flow problems. Namely we consider the volume, symmetry group, facets, vertices, face lattice, diameter, adjacency and incide ..."
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Cited by 17 (11 self)
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We survey and present new geometric and combinatorial propertiez of some polyhedra with application in combinatorial optimization, for example, the maxcut and multicommodity flow problems. Namely we consider the volume, symmetry group, facets, vertices, face lattice, diameter, adjacency and incidence relm:ons and connectivity of the metric polytope and its relatives. In partic~dar, using its large symmetry group, we completely describe all the 13 o:bits which form the 275 840 vertices of the 21dimensional metric polytope on 7 nodes and their incidence and adjacency relations. The edge connectivity, the/skeletons and a lifting procedure valid for a large class of vertices of the metric polytope are also given. Finally, we present an ordering of the facets of a polytope, based on their adjacency relations, for the enumeration of its vertices by the double description method.
The Structure of Circular Decomposable Metrics
 in: Proc. 4th ESA, Lect. Notes Comput. Sci. 1136 (SpringerVerlag
, 1996
"... Introduction Given a hard combinatorial optimization problem, a natural research direction is to find specially structured cases which can be solved more easily. These special cases may be useful in their own right or they may be used as a surrogate for finding approximate solutions in more general ..."
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Cited by 14 (1 self)
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Introduction Given a hard combinatorial optimization problem, a natural research direction is to find specially structured cases which can be solved more easily. These special cases may be useful in their own right or they may be used as a surrogate for finding approximate solutions in more general cases. There have been a number of special cases found for the Traveling Salesman Problem (TSP). In the TSP, there are n points and a distance function D[i; j] that maps pairs of points into nonnegative values representing the distances between the points. The objective is to find a permutation ß that minimizes P n i=1 D[ß(i); ß(i+ 1)] +D[ß(n); ß(0)]. We will work with symmetric distance functions, where D[i; j] = D[j;
The complexity of the consistency and Nrepresentability problems for quantum states
"... QMA (Quantum MerlinArthur) is the quantum analogue of the class NP. There are a few QMAcomplete problems, most of which are variants of the “Local Hamiltonian” problem introduced by Kitaev. In this dissertation we show some new QMAcomplete problems which are very different from those known previo ..."
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Cited by 8 (1 self)
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QMA (Quantum MerlinArthur) is the quantum analogue of the class NP. There are a few QMAcomplete problems, most of which are variants of the “Local Hamiltonian” problem introduced by Kitaev. In this dissertation we show some new QMAcomplete problems which are very different from those known previously, and have applications in quantum chemistry. The first one is “Consistency of Local Density Matrices”: given a collection of density matrices describing different subsets of an nqubit system (where each subset has constant size), decide whether these are consistent with some global state of all n qubits. This problem was first suggested by Aharonov. We show that it is QMAcomplete, via an oracle reduction from Local Hamiltonian. Our reduction is based on algorithms for convex optimization with a membership oracle, due to Yudin and Nemirovskii. Next we show that two problems from quantum chemistry, “Fermionic Local Hamiltonian” and “Nrepresentability, ” are QMAcomplete. These problems involve systems of fermions, rather than qubits; they arise in calculating the ground state energies of molecular systems. Nrepresentability is particularly interesting, as it is a key component
Monotone Maps, Sphericity and Bounded Second Eigenvalue
 Journal of Combinatorial Theory, Series B
, 2004
"... We consider monotone embeddings of a nite metric space into low dimensional normed space. That is, embeddings that respect the order among the distances in the original space. Our main interest is in embeddings into Euclidean spaces. We observe that any metric on n points can be embedded into l ..."
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Cited by 7 (0 self)
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We consider monotone embeddings of a nite metric space into low dimensional normed space. That is, embeddings that respect the order among the distances in the original space. Our main interest is in embeddings into Euclidean spaces. We observe that any metric on n points can be embedded into l 2 , while, (in a sense to be made precise later), for almost every npoint metric space, every monotone map must be into a space of dimension n) (Lemma 3).