Results 1  10
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51
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 238 (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.
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 47 (12 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.
Bidimensionality: New Connections between FPT Algorithms and PTASs
"... We demonstrate a new connection between fixedparameter tractability and approximation algorithms for combinatorial optimization problems on planar graphs and their generalizations. Specifically, we extend the theory of socalled “bidimensional” problems to show that essentially all such problems ha ..."
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Cited by 36 (5 self)
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We demonstrate a new connection between fixedparameter tractability and approximation algorithms for combinatorial optimization problems on planar graphs and their generalizations. Specifically, we extend the theory of socalled “bidimensional” problems to show that essentially all such problems have both subexponential fixedparameter algorithms and PTASs. Bidimensional problems include e.g. feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertexremoval problems, dominating set, edge dominating set, rdominating set, diameter, connected dominating set, connected edge dominating set, and connected rdominating set. We obtain PTASs for all of these problems in planar graphs and certain generalizations; of particular interest are our results for the two wellknown problems of connected dominating set and general feedback vertex set for planar graphs and their generalizations, for which PTASs were not known to exist. Our techniques generalize and in some sense unify the two main previous approaches for designing PTASs in planar graphs, namely, the LiptonTarjan separator approach [FOCS’77] and the Baker layerwise decomposition approach [FOCS’83]. In particular, we replace the notion of separators with a more powerful tool from the bidimensionality theory, enabling the first approach to apply to a much broader class of minimization problems than previously possible; and through the use of a structural backbone and thickening of layers we demonstrate how the second approach can be applied to problems with a “nonlocal” structure.
Object Location Using Path Separators
, 2006
"... We study a novel separator property called kpath separable. Roughly speaking, a kpath separable graph can be recursively separated into smaller components by sequentially removing k shortest paths. Our main result is that every minor free weighted graph is kpath separable. We then show that kpat ..."
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Cited by 35 (11 self)
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We study a novel separator property called kpath separable. Roughly speaking, a kpath separable graph can be recursively separated into smaller components by sequentially removing k shortest paths. Our main result is that every minor free weighted graph is kpath separable. We then show that kpath separable graphs can be used to solve several object location problems: (1) a smallworldization with an average polylogarithmic number of hops; (2) an (1 + ε)approximate distance labeling scheme with O(log n) space labels; (3) a stretch(1 + ε) compact routing scheme with tables of polylogarithmic space; (4) an (1+ε)approximate distance oracle with O(n log n) space and O(log n) query time. Our results generalizes to much wider classes of weighted graphs, namely to boundeddimension isometric sparable graphs.
Exact algorithms for treewidth and minimum fillin
 In Proceedings of the 31st International Colloquium on Automata, Languages and Programming (ICALP 2004). Lecture Notes in Comput. Sci
, 2004
"... We show that the treewidth and the minimum fillin of an nvertex graph can be computed in time O(1.8899 n). Our results are based on combinatorial proofs that an nvertex graph has O(1.7087 n) minimal separators and O(1.8135 n) potential maximal cliques. We also show that for the class of ATfree g ..."
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Cited by 26 (15 self)
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We show that the treewidth and the minimum fillin of an nvertex graph can be computed in time O(1.8899 n). Our results are based on combinatorial proofs that an nvertex graph has O(1.7087 n) minimal separators and O(1.8135 n) potential maximal cliques. We also show that for the class of ATfree graphs the running time of our algorithms can be reduced to O(1.4142 n).
Advances in metric embedding theory
 IN STOC ’06: PROCEEDINGS OF THE THIRTYEIGHTH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING
, 2006
"... Metric Embedding plays an important role in a vast range of application areas such as computer vision, computational biology, machine learning, networking, statistics, and mathematical psychology, to name a few. The theory of metric embedding received much attention in recent years by mathematicians ..."
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Cited by 26 (8 self)
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Metric Embedding plays an important role in a vast range of application areas such as computer vision, computational biology, machine learning, networking, statistics, and mathematical psychology, to name a few. The theory of metric embedding received much attention in recent years by mathematicians as well as computer scientists and has been applied in many algorithmic applications. A cornerstone of the field is a celebrated theorem of Bourgain which states that every finite metric space on n points embeds in Euclidean space with O(log n) distortion. Bourgain’s result is best possible when considering the worst case distortion over all pairs of points in the metric space. Yet, it is possible that an embedding can do much better in terms of the average distortion. Indeed, in most practical applications of metric embedding the main criteria for the quality of an embedding is its average distortion over all pairs. In this paper we provide an embedding with constant average distortion for arbitrary metric spaces, while maintaining the same worst case bound provided by Bourgain’s theorem. In fact, our embedding possesses a much stronger property. We define the ℓqdistortion of a uniformly distributed pair of points. Our embedding achieves the best possible ℓqdistortion for all 1 ≤ q ≤ ∞ simultaneously. These results have several algorithmic implications, e.g. an O(1) approximation for the unweighted uncapacitated quadratic assignment problem. The results are based on novel embedding methods which improve on previous methods in another important aspect: the dimension. The dimension of an embedding is of very high importance in particular in applications and much effort has been invested in analyzing it. However, no previous result im
Linearity of Grid Minors in Treewidth with Applications through Bidimensionality
, 2005
"... We prove that any Hminorfree graph, for a fixed graph H, of treewidth w has an \Omega (w) *\Omega ( w) grid graph as a minor. Thus grid minors suffice to certify that Hminorfree graphs havelarge treewidth, up to constant factors. This strong relationship was previously known for the special cas ..."
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Cited by 21 (1 self)
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We prove that any Hminorfree graph, for a fixed graph H, of treewidth w has an \Omega (w) *\Omega ( w) grid graph as a minor. Thus grid minors suffice to certify that Hminorfree graphs havelarge treewidth, up to constant factors. This strong relationship was previously known for the special cases of planar graphs and boundedgenus graphs, and is known not to hold for generalgraphs. The approach of this paper can be viewed more generally as a framework for extending combinatorial results on planar graphs to hold on Hminorfree graphs for any fixed H. Ourresult has many combinatorial consequences on bidimensionality theory, parametertreewidth bounds, separator theorems, and bounded local treewidth; each of these combinatorial resultshas several algorithmic consequences including subexponential fixedparameter algorithms and approximation algorithms.
Subexponential parameterized algorithms
 Computer Science Review
"... We give a review of a series of techniques and results on the design of subexponential parameterized algorithms for graph problems. The design of such algorithms usually consists of two main steps: first find a branch (or tree) decomposition of the input graph whose width is bounded by a sublinear ..."
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Cited by 18 (8 self)
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We give a review of a series of techniques and results on the design of subexponential parameterized algorithms for graph problems. The design of such algorithms usually consists of two main steps: first find a branch (or tree) decomposition of the input graph whose width is bounded by a sublinear function of the parameter and, second, use this decomposition to solve the problem in time that is single exponential to this bound. The main tool for the first step is Bidimensionality Theory. Here we present the potential, but also the boundaries, of this theory. For the second step, we describe recent techniques, associating the analysis of subexponential algorithms to combinatorial bounds related to Catalan numbers. As a result, we have 2 O( √ k) · n O(1) time algorithms for a wide variety of parameterized problems on graphs, where n is the size of the graph and k is the parameter. 1
Nondeterministic Graph Searching: From Pathwidth to Treewidth
 In 30th International Symposium on Mathematical Foundations of Computer Science (MFCS), LNCS 3618
, 2005
"... Abstract. We introduce nondeterministic graph searching with a controlled amount of nondeterminism and show how this new tool can be used in algorithm design and combinatorial analysis applying to both pathwidth and treewidth. We prove equivalence between this gametheoretic approach and graph decom ..."
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Cited by 17 (6 self)
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Abstract. We introduce nondeterministic graph searching with a controlled amount of nondeterminism and show how this new tool can be used in algorithm design and combinatorial analysis applying to both pathwidth and treewidth. We prove equivalence between this gametheoretic approach and graph decompositions called qbranched tree decompositions, which can be interpreted as a parameterized version of tree decompositions. Path decomposition and (standard) tree decomposition are two extreme cases of qbranched tree decompositions. The equivalence between nondeterministic graph searching and qbranched tree decomposition enables us to design an exact (exponential time) algorithm computing qbranched treewidth for all q ≥ 0, which is thus valid for both treewidth and pathwidth. This algorithm performs as fast as the best known exact algorithm for pathwidth. Conversely, this equivalence also enables us to design a lower bound on the amount of nondeterminism required to search a graph with the minimum number of searchers.