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112
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 r ..."
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Cited by 448 (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
A Tight Bound on Approximating Arbitrary Metrics by Tree Metrics
 In Proceedings of the 35th Annual ACM Symposium on Theory of Computing
, 2003
"... In this paper, we show that any n point metric space can be embedded into a distribution over dominating tree metrics such that the expected stretch of any edge is O(log n). This improves upon the result of Bartal who gave a bound of O(log n log log n). Moreover, our result is existentially tight; t ..."
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Cited by 260 (7 self)
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In this paper, we show that any n point metric space can be embedded into a distribution over dominating tree metrics such that the expected stretch of any edge is O(log n). This improves upon the result of Bartal who gave a bound of O(log n log log n). Moreover, our result is existentially tight; there exist metric spaces where any tree embedding must have distortion#sto n)distortion. This problem lies at the heart of numerous approximation and online algorithms including ones for group Steiner tree, metric labeling, buyatbulk network design and metrical task system. Our result improves the performance guarantees for all of these problems.
Expander Flows, Geometric Embeddings and Graph Partitioning
 IN 36TH ANNUAL SYMPOSIUM ON THE THEORY OF COMPUTING
, 2004
"... We give a O( log n)approximation algorithm for sparsest cut, balanced separator, and graph conductance problems. This improves the O(log n)approximation of Leighton and Rao (1988). We use a wellknown semidefinite relaxation with triangle inequality constraints. Central to our analysis is a ..."
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Cited by 235 (18 self)
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We give a O( log n)approximation algorithm for sparsest cut, balanced separator, and graph conductance problems. This improves the O(log n)approximation of Leighton and Rao (1988). We use a wellknown semidefinite relaxation with triangle inequality constraints. Central to our analysis is a geometric theorem about projections of point sets in , whose proof makes essential use of a phenomenon called measure concentration.
Approximation Algorithms for Disjoint Paths Problems
, 1996
"... The construction of disjoint paths in a network is a basic issue in combinatorial optimization: given a network, and specified pairs of nodes in it, we are interested in finding disjoint paths between as many of these pairs as possible. This leads to a variety of classical NPcomplete problems for w ..."
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Cited by 139 (0 self)
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The construction of disjoint paths in a network is a basic issue in combinatorial optimization: given a network, and specified pairs of nodes in it, we are interested in finding disjoint paths between as many of these pairs as possible. This leads to a variety of classical NPcomplete problems for which very little is known from the point of view of approximation algorithms. It has recently been brought into focus in work on problems such as VLSI layout and routing in highspeed networks; in these settings, the current lack of understanding of the disjoint paths problem is often an obstacle to the design of practical heuristics.
The Complexity of Multiterminal Cuts
 SIAM Journal on Computing
, 1994
"... In the Multiterminal Cut problem we are given an edgeweighted graph and a subset of the vertices called terminals, and asked for a minimum weight set of edges that separates each terminal from all the others. When the number k of terminals is two, this is simply the mincut, maxflow problem, and ..."
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Cited by 139 (0 self)
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In the Multiterminal Cut problem we are given an edgeweighted graph and a subset of the vertices called terminals, and asked for a minimum weight set of edges that separates each terminal from all the others. When the number k of terminals is two, this is simply the mincut, maxflow problem, and can be solved in polynomial time. We show that the problem becomes NPhard as soon as k = 3, but can be solved in polynomial time for planar graphs for any fixed k. The planar problem is NPhard, however, if k is not fixed. We also describe a simple approximation algorithm for arbitrary graphs that is guaranteed to come within a factor of 2  2/k of the optimal cut weight.
An O(log k) approximate mincut maxflow theorem and approximation algorithm
 SIAM J. Comput
, 1998
"... Abstract. 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. A ..."
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Cited by 124 (6 self)
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Abstract. 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.
THE PRIMALDUAL METHOD FOR APPROXIMATION ALGORITHMS AND ITS APPLICATION TO NETWORK DESIGN PROBLEMS
"... The primaldual method is a standard tool in the design of algorithms for combinatorial optimization problems. This chapter shows how the primaldual method can be modified to provide good approximation algorithms for a wide variety of NPhard problems. We concentrate on results from recent researc ..."
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Cited by 119 (7 self)
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The primaldual method is a standard tool in the design of algorithms for combinatorial optimization problems. This chapter shows how the primaldual method can be modified to provide good approximation algorithms for a wide variety of NPhard problems. We concentrate on results from recent research applying the primaldual method to problems in network design.
Excluded Minors, Network Decomposition, and Multicommodity Flow
, 1993
"... In this paper we show that, given a graph and parameters ffi and r, we can find either a Kr;r minor or an edgecut of size O(mr=ffi) whose removal yields components of weak diameter O(r 2 ffi); i.e., every pair of nodes in such a component are at distance O(r 2 ffi) in the original graph. Usi ..."
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Cited by 104 (6 self)
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In this paper we show that, given a graph and parameters ffi and r, we can find either a Kr;r minor or an edgecut of size O(mr=ffi) whose removal yields components of weak diameter O(r 2 ffi); i.e., every pair of nodes in such a component are at distance O(r 2 ffi) in the original graph. Using this lemma, we improve the best known bounds for the mincut maxflow ratio for multicommodity flows in graphs with forbidden small minors. In general graphs, it was known that the ratio is O(log k) for the uniformdemand case (the case where there is a unitdemand commodity between every pair of nodes), and that the ratio is O(log 2 k) for arbitrary demands, where k is the number of commodities. In this paper we show that for graphs excluding any fixed graph as a minor (e.g. planar graphs or boundedgenus graphs), the ratio is O(1) for the uniformdemand case and O(log k) for the arbitrary demand case. For such graphs, our method yields minratio cut approximation algorithms wit...
Hardness Of Approximations
, 1996
"... This chapter is a selfcontained survey of recent results about the hardness of approximating NPhard optimization problems. ..."
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Cited by 101 (4 self)
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This chapter is a selfcontained survey of recent results about the hardness of approximating NPhard optimization problems.
DivideandConquer Approximation Algorithms via Spreading Metrics
, 1996
"... We present a novel divideandconquer paradigm for approximating NPhard graph optimization problems. The paradigm models graph optimization problems that satisfy two properties: First, a divideandconquer approach is applicable. Second, a fractional spreading metric is computable in polynomial tim ..."
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Cited by 97 (10 self)
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We present a novel divideandconquer paradigm for approximating NPhard graph optimization problems. The paradigm models graph optimization problems that satisfy two properties: First, a divideandconquer approach is applicable. Second, a fractional spreading metric is computable in polynomial time. The spreading metric assigns rational lengths to either edges or vertices of the input graph, such that all subgraphs on which the optimization problem is nontrivial have large diameters. In addition, the spreading metric provides a lower bound, ø , on the cost of solving the optimization problem. We present a polynomial time approximation algorithm for problems modeled by our paradigm whose approximation factor is O (minflog ø log log ø; log k log log kg), where k denotes the number of "interesting" vertices in the problem instance, and is at most the number of vertices. We present seven problems that can be formulated to fit the paradigm. For all these problems our algorithm improves ...