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22
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
Multicommodity maxflow mincut theorems and their use in designing approximation algorithms
 J. ACM
, 1999
"... In this paper, we establish maxflow mincut theorems for several important classes of multicommodity flow problems. In particular, we show that for any nnode multicommodity flow problem with uniform demands, the maxflow for the problem is within an O(log n) factor of the upper bound implied by ..."
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Cited by 357 (6 self)
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In this paper, we establish maxflow mincut theorems for several important classes of multicommodity flow problems. In particular, we show that for any nnode multicommodity flow problem with uniform demands, the maxflow for the problem is within an O(log n) factor of the upper bound implied by the mincut. The result (which is existentially optimal) establishes an important analogue of the famous 1commodity maxflow mincut theorem for problems with multiple commodities. The result also has substantial applications to the field of approximation algorithms. For example, we use the flow result to design the first polynomialtime (polylog ntimesoptimal) approximation algorithms for wellknown NPhard optimization problems such as graph partitioning, mincut linear arrangement, crossing number, VLSI layout, and minimum feedback arc set. Applications of the flow results to path routing problems, network reconfiguration, communication in distributed networks, scientific computing and rapidly mixing Markov chains are also described in the paper.
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 166 (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.
Approximate MaxFlow Min(multi)cut Theorems and Their Applications
 SIAM Journal on Computing
, 1993
"... Consider the multicommodity flow problem in which the object is to maximize the sum of commodities routed. We prove the following approximate maxflow minmulticut theorem: min multicut O(logk) max flow min multicut; where k is the number of commodities. Our proof is constructive; it enables us ..."
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Cited by 160 (3 self)
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Consider the multicommodity flow problem in which the object is to maximize the sum of commodities routed. We prove the following approximate maxflow minmulticut theorem: min multicut O(logk) max flow min multicut; where k is the number of commodities. Our proof is constructive; it enables us to find a multicut within O(log k) of the max flow (and hence also the optimal multicut). In addition, the proof technique provides a unified framework in which one can also analyse the case of flows with specified demands, of LeightonRao and Klein et.al., and thereby obtain an improved bound for the latter problem. 1 Introduction Much of flow theory, and the theory of cuts in graphs, is built around a single theorem  the celebrated maxflow mincut theorem of Ford and Fulkerson [FF], and Elias, Feinstein and Shannon [EFS]. The power of this theorem lies in that it relates two fundamental graphtheoretic entities via the potent mechanism of a minmax relation. The importance of this theor...
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.
A Bundle Type DualAscent Approach to Linear Multicommodity MinCost Flow Problems
"... We present a Cost Decomposition approach for the linear Multicommodity MinCost Flow problem, where the mutual capacity constraints are dualized and the resulting Lagrangean Dual is solved with a dualascent algorithm belonging to the class of Bundle methods. Although decomposition approaches to bl ..."
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Cited by 36 (16 self)
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We present a Cost Decomposition approach for the linear Multicommodity MinCost Flow problem, where the mutual capacity constraints are dualized and the resulting Lagrangean Dual is solved with a dualascent algorithm belonging to the class of Bundle methods. Although decomposition approaches to blockstructured Linear Programs have been reported not to be competitive with generalpurpose software, our extensive computational comparison shows that, when carefully implemented, a decomposition algorithm can outperform several other approaches, especially on problems where the number of commodities is “large ” with respect to the size of the graph. Our specialized Bundle algorithm is characterized by a new heuristic for the trust region parameter handling, and embeds a specialized Quadratic Program solver that allows the efficient implementation of strategies for reducing the number of active Lagrangean variables. We also exploit the structural properties of the singlecommodity MinCost Flow subproblems to reduce the overall computational cost. The proposed approach can be easily extended to handle variants of the problem.
Approximation Algorithms for Steiner and Directed Multicuts
 JOURNAL OF ALGORITHMS
, 1996
"... In this paper we consider the steiner multicut problem. This is a generalization of the minimum multicut problem where instead of separating node pairs, the goal is to find a minimum weight set of edges that separates all given sets of nodes. A set is considered separated if it is not contained in ..."
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Cited by 27 (1 self)
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In this paper we consider the steiner multicut problem. This is a generalization of the minimum multicut problem where instead of separating node pairs, the goal is to find a minimum weight set of edges that separates all given sets of nodes. A set is considered separated if it is not contained in a single connected component. We show an O(log 3 (kt)) approximation algorithm for the steiner multicut problem, where k is the number of sets and t is the maximum cardinality of a set. This improves the O(t log k) bound that easily follows from the previously known multicut results. We also consider an extension of multicuts to directed case, namely the problem of finding a minimumweight set of edges whose removal ensures that none of the strongly connected components includes one of the prespecified k node pairs. In this paper we describe an O(log 2 k) approximation algorithm for this directed multicut problem. If k n, this represents and an improvement over the O(logn log ...
Multicuts in Unweighted Graphs and Digraphs with Bounded Degree and Bounded TreeWidth
, 1998
"... this paper. Also, we show that Directed Edge Multicut is NPhard in digraphs with treewidth one and maximum in and out degree three. Other hardness results indicate why we cannot eliminate any of the three restrictionsunweighted, bounded degree and bounded treewidthon the input graph and sti ..."
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Cited by 24 (0 self)
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this paper. Also, we show that Directed Edge Multicut is NPhard in digraphs with treewidth one and maximum in and out degree three. Other hardness results indicate why we cannot eliminate any of the three restrictionsunweighted, bounded degree and bounded treewidthon the input graph and still obtain a PTAS. It is known [1] that for a Max SNPhard problem, unless P=NP, no PTAS exists. We have already seen that Unweighted Edge Multicut is Max SNPhard in stars [9], so letting the input graph have unbounded degree makes the problem harder. We show that Weighted Edge Multicut is Max SNPhard in binary trees, therefore letting the input graph be weighted makes the problem harder. Finally, we show that Unweighted Edge Multicut is Max SNPhard if the input graphs are walls. Walls, to be formally defined in Section 6, have degree at most three and unbounded treewidth. We conclude that letting the input graph have unbounded treewidth makes the problem significantly harder
Bounds on the MaxFlow MinCut Ratio for Directed Multicommodity Flows
, 1993
"... The most wellknown theorem in combinatorial optimization is the classical maxflow mincut theorem of Ford and Fulkerson. This theorem serves as the basis for deriving efficient algorithms for finding maxflows and mincuts. Starting with the work of Leighton and Rao, significant effort was directe ..."
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Cited by 8 (3 self)
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The most wellknown theorem in combinatorial optimization is the classical maxflow mincut theorem of Ford and Fulkerson. This theorem serves as the basis for deriving efficient algorithms for finding maxflows and mincuts. Starting with the work of Leighton and Rao, significant effort was directed towards finding approximate analogs for the undirected multicommodity flow problem. In this paper we consider an approximate maxflow mincut theorem for directed graphs. We prove a polylogarithmic bound on the worst case ratio between the minimum multicut and the value of the maximum multicommodity flow in the special case when the demands are symmetric. The method presented in this paper can be used to give polynomial time polylogarithmic approximation algorithms for the corresponding minimum directed multicut problems. The problem with symmetric demands extends the only previously known special case concerning directed graphs due to Leighton and Rao, who proved an O(log n) bound for th...