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17
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 322 (8 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.
Algorithmic Graph Minor Theory: Decomposition, Approximation, and Coloring
 In 46th Annual IEEE Symposium on Foundations of Computer Science
, 2005
"... At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful structural theorem capturing the structure of graphs excluding a fixed minor. This result is used throughout graph theory and graph algorithms, but is existential. We develop a polynomialtime algorithm using topolog ..."
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Cited by 61 (17 self)
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At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful structural theorem capturing the structure of graphs excluding a fixed minor. This result is used throughout graph theory and graph algorithms, but is existential. We develop a polynomialtime algorithm using topological graph theory to decompose a graph into the structure guaranteed by the theorem: a cliquesum of pieces almostembeddable into boundedgenus surfaces. This result has many applications. In particular, we show applications to developing many approximation algorithms, including a 2approximation to graph coloring, constantfactor approximations to treewidth and the largest grid minor, combinatorial polylogarithmicapproximation to halfintegral multicommodity flow, subexponential fixedparameter algorithms, and PTASs for many minimization and maximization problems, on graphs excluding a fixed minor. 1.
Equivalence of Local Treewidth and Linear Local Treewidth and its Algorithmic Applications
 In Proceedings of the 15th ACMSIAM Symposium on Discrete Algorithms (SODA’04
, 2003
"... We solve an open problem posed by Eppstein in 1995 [14, 15] and reenforced by Grohe [16, 17] concerning locally bounded treewidth in minorclosed families of graphs. A graph has bounded local treewidth if the subgraph induced by vertices within distance r of any vertex has treewidth bounded by a f ..."
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Cited by 32 (11 self)
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We solve an open problem posed by Eppstein in 1995 [14, 15] and reenforced by Grohe [16, 17] concerning locally bounded treewidth in minorclosed families of graphs. A graph has bounded local treewidth if the subgraph induced by vertices within distance r of any vertex has treewidth bounded by a function of r (not n). Eppstein characterized minorclosed families of graphs with bounded local treewidth as precisely minorclosed families that minorexclude an apex graph, where an apex graph has one vertex whose removal leaves a planar graph. In particular, Eppstein showed that all apexminorfree graphs have bounded local treewidth, but his bound is doubly exponential in r, leaving open whether a tighter bound could be obtained. We improve this doubly exponential bound to a linear bound, which is optimal. In particular, any minorclosed graph family with bounded local treewidth has linear local treewidth. Our bound generalizes previously known linear bounds for special classes of graphs proved by several authors. As a consequence of our result, we obtain substantially faster polynomialtime approximation schemes for a broad class of problems in apexminorfree graphs, improving the running time from .
A simpler algorithm and shorter proof for the graph minor decomposition
 In Proceedings of the 43rd ACM Symposium on Theory of Computing
, 2011
"... At the core of the RobertsonSeymour theory of graph minors lies a powerful decomposition theorem which captures, for any fixed graph H, the common structural features of all the graphs which do not contain H as a minor. Robertson and Seymour used this result to prove Wagner’s Conjecture that finit ..."
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Cited by 12 (3 self)
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At the core of the RobertsonSeymour theory of graph minors lies a powerful decomposition theorem which captures, for any fixed graph H, the common structural features of all the graphs which do not contain H as a minor. Robertson and Seymour used this result to prove Wagner’s Conjecture that finite graphs are wellquasiordered under the graph minor relation, as well as give a polynomial time algorithm for the disjoint paths problem when the number of the terminals is fixed. The theorem has since found numerous applications, both in graph theory and theoretical computer science. The original proof runs more than 400 pages and the techniques used are highly nontrivial. In this paper, we give a simplified algorithm for finding the decomposition based on a new constructive proof of the decomposition theorem for graphs excluding a fixed minor H. The new proof is both dramatically simpler and shorter, making these results and techniques more accessible. The algorithm runs in time O(n3), as does the original algorithm of Robertson and Seymour. Moreover, our proof gives an explicit bound on the constants in the O notation, whereas the original proof of Robertson and Seymour does not. Categories and Subject Descriptors G.2.2 [Discrete Mathematics]: Graph Theory—graph algorithms, path and circuit problems
FlowCut Gaps for Integer and Fractional Multiflows
, 2009
"... Consider a routing problem instance consisting of a demand graph H = (V, E(H)) and a supply graph G = (V, E(G)). If the pair obeys the cut condition, then the flowcut gap for this instance is the minimum value C such that there exists a feasible multiflow for H if each edge of G is given capacity C ..."
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Cited by 10 (1 self)
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Consider a routing problem instance consisting of a demand graph H = (V, E(H)) and a supply graph G = (V, E(G)). If the pair obeys the cut condition, then the flowcut gap for this instance is the minimum value C such that there exists a feasible multiflow for H if each edge of G is given capacity C. It is wellknown that the flowcut gap may be greater than 1 even in the case where G is the (seriesparallel) graph K2,3. In this paper we are primarily interested in the “integer ” flowcut gap. What is the minimum value C such that there exists a feasible integer valued multiflow for H if each edge of G is given capacity C? We formulate a conjecture that states that the integer flowcut gap is quantitatively related to the fractional flowcut gap. In particular this strengthens the wellknown conjecture that the flowcut gap in planar and minorfree graphs is O(1) [12] to suggest that the integer flowcut gap is O(1). We give several technical tools and results on nontrivial special classes of graphs to give evidence for the conjecture and further explore the “primal ” method for understanding flowcut gaps; this is in contrast to and orthogonal to the highly successful metric embeddings approach. Our results include the following: • Let G be obtained by seriesparallel operations starting from an edge st, and consider orienting all edges in G in the direction from s to t. A demand is compliant if its endpoints are joined by a directed
Improved embeddings of graph metrics into random trees
 In Proceedings of the 17th Annual ACMSIAM Symposium on Discrete Algorithms (To appear
, 2006
"... Over the past decade, numerous algorithms have been developed using the fact that the distances in any npoint metric (V, d) can be approximated to within O(log n) by distributions D over trees on the point set V [3, 9]. However, when the metric (V, d) is the shortestpath metric of an edge weighted ..."
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Cited by 6 (0 self)
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Over the past decade, numerous algorithms have been developed using the fact that the distances in any npoint metric (V, d) can be approximated to within O(log n) by distributions D over trees on the point set V [3, 9]. However, when the metric (V, d) is the shortestpath metric of an edge weighted graph G = (V, E), a natural requirement is to obtain such a result where the support of the distribution D is only over subtrees of G. For a long time, the best result satisfying this stronger requirement was a exp { √ log n log log n} distortion result of Alon et al. [1]. In a recent breakthrough, Elkin et al. [8] improved the distortion to O(log 2 n log log n). (The best lower bound on the distortion is Ω(log n), say, for the nvertex grid [1].) In this paper, we give a construction that improves the distortion to O(log 2 n), improving slightly on the EEST construction. The main contribution of this paper is in the analysis: we use an algorithm which is similar to one used by EEST to give a distortion of O(log 3 n), but using a new probabilistic analysis, we eliminate one of the logarithmic factors. The ideas and techniques we use to obtain this logarithmic improvement seem orthogonal to those used earlier in such situations—e.g., Seymour’s decomposition scheme [4, 8] or the cutting procedures of CKR/FRT [5, 9], both which do not seem to give a guarantee of better than O(log 2 n log log n) for this problem. We hope that our ideas (perhaps in conjunction with some of these others) will ultimately lead to an O(log n) distortion embedding of graph metrics into distributions over their spanning trees. 1
Lower bounds for embedding into distributions over excluded minor graph families
 Lecture Notes in Computer Science (proceedings of the 12th European Symposium on Algorithms
, 2004
"... It was shown recently by Fakcharoenphol et al. [9] that arbitrary finite metrics can be embedded into distributions over tree metrics with distortion O(log n). It is also known that this bound is tight since there are expander graphs which cannot be embedded into distributions over trees with better ..."
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Cited by 6 (0 self)
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It was shown recently by Fakcharoenphol et al. [9] that arbitrary finite metrics can be embedded into distributions over tree metrics with distortion O(log n). It is also known that this bound is tight since there are expander graphs which cannot be embedded into distributions over trees with better than Ω(log n) distortion. We show that this same lower bound holds for embeddings into distributions over any minor excluded family. Given a family of graphs F which excludes minor M where M  = k, we explicitly construct a family of graphs with treewidth(k + 1) which cannot be embedded into a distribution over F with better than Ω(log n) distortion. Thus, while these minor excluded families of graphs are more expressive than trees, they do not provide asymptotically better approximations in general. An important corollary of this is that graphs of treewidthk cannot be embedded into distributions over graphs of treewidth(k−3) with distortion less than Ω(log n). We also extend a result of Alon et al. [1] by showing that for any k, planar graphs cannot be embedded into distributions over treewidthk graphs with better than Ω(log n) distortion. 1
Multicommodity Flows and Cuts in Polymatroidal Networks
, 2011
"... We consider multicommodity flow and cut problems in polymatroidal networks where there are submodular capacity constraints on the edges incident to a node. Polymatroidal networks were introduced by Lawler and Martel [24] and Hassin [18] in the singlecommodity setting and are closely related to the ..."
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Cited by 6 (2 self)
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We consider multicommodity flow and cut problems in polymatroidal networks where there are submodular capacity constraints on the edges incident to a node. Polymatroidal networks were introduced by Lawler and Martel [24] and Hassin [18] in the singlecommodity setting and are closely related to the submodular flow model of Edmonds and Giles [10]; the wellknown maxflowmincut theorem generalizes to this more general setting. Polymatroidal networks for the multicommodity case have not, as far as the authors are aware, been previously explored. Our work is primarily motivated by applications to information flow in wireless networks. We also consider the notion of undirected polymatroidal networks and observe that they provide a natural way to generalize flows and cuts in edge and node capacitated undirected networks. We establish polylogarithmic flowcut gap results in several scenarios that have been previously considered in the standard network flow models where capacities are on the edges or nodes [25, 26, 14, 23, 13]. Our results from a preliminary version have already found applications in wireless network information flow [20, 21] and we anticipate more in the future. On the technical side our key tools are the formulation and analysis of the dual of the flow relaxations via continuous extensions of submodular functions, in particular the Lovász extension. For directed graphs we rely on a simple yet useful reduction from
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.
Preserving Terminal Distances using Minors ⋆
"... Abstract. We introduce the following notion of compressing an undirected graph G with (nonnegative) edgelengths and terminal vertices R ⊆ V (G). A distancepreserving minor is a minor G ′ (of G) with possibly different edgelengths, such that R ⊆ V (G ′ ) and the shortestpath distance between ever ..."
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Cited by 4 (3 self)
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Abstract. We introduce the following notion of compressing an undirected graph G with (nonnegative) edgelengths and terminal vertices R ⊆ V (G). A distancepreserving minor is a minor G ′ (of G) with possibly different edgelengths, such that R ⊆ V (G ′ ) and the shortestpath distance between every pair of terminals is exactly the same in G and in G ′. We ask: what is the smallest f ∗ (k) such that every graph G with k = R  terminals admits a distancepreserving minor G ′ with at most f ∗ (k) vertices? Simple analysis shows that f ∗ (k) ≤ O(k 4). Our main result proves that f ∗ (k) ≥ Ω(k 2), significantly improving over the trivial f ∗ (k) ≥ k. Our lower bound holds even for planar graphs G, in contrast to graphs G of constant treewidth, for which we prove that O(k) vertices suffice. 1