Results 1 
9 of
9
Subexponential parameterized algorithms on graphs of boundedgenus and Hminorfree Graphs
"... ... Building on these results, we develop subexponential fixedparameter algorithms for dominating set, vertex cover, and set cover in any class of graphs excluding a fixed graph H as a minor. Inparticular, this general category of graphs includes planar graphs, boundedgenus graphs, singlecrossing ..."
Abstract

Cited by 63 (22 self)
 Add to MetaCart
... Building on these results, we develop subexponential fixedparameter algorithms for dominating set, vertex cover, and set cover in any class of graphs excluding a fixed graph H as a minor. Inparticular, this general category of graphs includes planar graphs, boundedgenus graphs, singlecrossingminorfree graphs, and anyclass of graphs that is closed under taking minors. Specifically, the running time is 2O(pk)nh, where h is a constant depending onlyon H, which is polynomial for k = O(log² n). We introducea general approach for developing algorithms on Hminorfreegraphs, based on structural results about Hminorfree graphs at the
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 ..."
Abstract

Cited by 56 (15 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 31 (10 self)
 Add to MetaCart
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 .
XML stream processing using treeedit distance embeddings
 ACM Trans. on Database Systems
, 2005
"... We propose the first known solution to the problem of correlating, in small space, continuous streams of XML data through approximate (structure and content) matching, as defined by a general treeedit distance metric. The key element of our solution is a novel algorithm for obliviously embedding tr ..."
Abstract

Cited by 14 (0 self)
 Add to MetaCart
We propose the first known solution to the problem of correlating, in small space, continuous streams of XML data through approximate (structure and content) matching, as defined by a general treeedit distance metric. The key element of our solution is a novel algorithm for obliviously embedding treeedit distance metrics into an L1 vector space while guaranteeing a (worstcase) upper bound of O(log 2 n log ∗ n)onthe distance distortion between any data trees with at most n nodes. We demonstrate how our embedding algorithm can be applied in conjunction with known random sketching techniques to (1) build a compact synopsis of a massive, streaming XML data tree that can be used as a concise surrogate for the full tree in approximate treeedit distance computations; and (2) approximate the result of treeeditdistance similarity joins over continuous XML document streams. Experimental results from an empirical study with both synthetic and reallife XML data trees validate our approach, demonstrating that the averagecase behavior of our embedding techniques is much better than what would be predicted from our theoretical worstcase distortion bounds. To the best of our knowledge, these are the first algorithmic results on lowdistortion embeddings for treeedit distance metrics, and on correlating (e.g., through similarity joins) XML data in the streaming model. Categories and Subject Descriptors: H.2.4 [Database Management]: Systems—Query processing; G.2.1 [Discrete Mathematics]: Combinatorics—Combinatorial algorithms
Near Linear Lower Bound for Dimension Reduction in ℓ1
"... Abstract — Given a set of n points in ℓ1, how many dimensions are needed to represent all pairwise distances within a specific distortion? This dimensiondistortion tradeoff question is well understood for the ℓ2 norm, where O((log n)/ɛ 2) dimensions suffice to achieve 1 + ɛ distortion. In sharp con ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
(Show Context)
Abstract — Given a set of n points in ℓ1, how many dimensions are needed to represent all pairwise distances within a specific distortion? This dimensiondistortion tradeoff question is well understood for the ℓ2 norm, where O((log n)/ɛ 2) dimensions suffice to achieve 1 + ɛ distortion. In sharp contrast, there is a significant gap between upper and lower bounds for dimension reduction in ℓ1. A recent result shows that distortion 1 + ɛ can be achieved with n/ɛ 2 dimensions. On the other hand, the only lower bounds known are that distortion δ requires n Ω(1/δ2) dimensions and that distortion 1+ɛ requires n 1/2−O(ɛ log(1/ɛ)) dimensions. In this work, we show the first near linear lower bounds for dimension reduction in ℓ1. In particular, we show that 1 + ɛ distortion requires at least n 1−O(1 / log(1/ɛ)) dimensions. Our proofs are combinatorial, but inspired by linear programming. In fact, our techniques lead to a simple combinatorial argument that is equivalent to the LP based proof of BrinkmanCharikar for lower bounds on dimension reduction in ℓ1. Keywordsdimension reduction, metric embedding 1.
Approximating the listchromatic number and the chromatic number in minorclosed and oddminorclosed classes of graphs
"... ..."
(Show Context)
An algorithm for 1 nearest neighbor search via monotonic embedding
"... Abstract Fast algorithms for nearest neighbor (NN) search have in large part focused on 2 distance. Here we develop an approach for 1 distance that begins with an explicit and exactly distancepreserving embedding of the points into 2 2 . We show how this can efficiently be combined with randompro ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract Fast algorithms for nearest neighbor (NN) search have in large part focused on 2 distance. Here we develop an approach for 1 distance that begins with an explicit and exactly distancepreserving embedding of the points into 2 2 . We show how this can efficiently be combined with randomprojection based methods for 2 NN search, such as localitysensitive hashing (LSH) or random projection trees. We rigorously establish the correctness of the methodology and show by experimentation using LSH that it is competitive in practice with available alternatives.