Results 1  10
of
17
Approximation algorithms via contraction decomposition
 Proc. 18th Ann. ACMSIAM Symp. Discrete Algorithms ACMSIAM symposium on Discrete algorithms
, 2007
"... We prove that the edges of every graph of bounded (Euler) genus can be partitioned into any prescribed number k of pieces such that contracting any piece results in a graph of bounded treewidth (where the bound depends on k). This decomposition result parallels an analogous, simpler result for edge ..."
Abstract

Cited by 28 (7 self)
 Add to MetaCart
We prove that the edges of every graph of bounded (Euler) genus can be partitioned into any prescribed number k of pieces such that contracting any piece results in a graph of bounded treewidth (where the bound depends on k). This decomposition result parallels an analogous, simpler result for edge deletions instead of contractions, obtained in [Bak94, Epp00, DDO + 04, DHK05], and it generalizes a similar result for “compression ” (a variant of contraction) in planar graphs [Kle05]. Our decomposition result is a powerful tool for obtaining PTASs for contractionclosed problems (whose optimal solution only improves under contraction), a much more general class than minorclosed problems. We prove that any contractionclosed problem satisfying just a few simple conditions has a PTAS in boundedgenus graphs. In particular, our framework yields PTASs for the weighted Traveling Salesman Problem and for minimumweight cedgeconnected submultigraph on boundedgenus graphs, improving and generalizing previous algorithms of [GKP95, AGK + 98, Kle05, Gri00, CGSZ04, BCGZ05]. We also highlight the only main difficulty in extending our results to general Hminorfree graphs.
Expander graphs, gonality and variation of Galois representations, arXiv:1008.3675 [EMV
 Duke Math. J
"... Abstract. We show that families of coverings of an algebraic curve where the associated CayleySchreier graphs form an expander family exhibit strong forms of geometric growth. Combining this general result with finiteness statements for rational points under such conditions, we derive results conce ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
(Show Context)
Abstract. We show that families of coverings of an algebraic curve where the associated CayleySchreier graphs form an expander family exhibit strong forms of geometric growth. Combining this general result with finiteness statements for rational points under such conditions, we derive results concerning the variation of Galois representations in oneparameter families of abelian varieties. 1.
Eigenvalue bounds, spectral partitioning, and metrical deformations via flows
"... ..."
(Show Context)
On the blackbox complexity of Sperner’s Lemma
 In FCT 2005
, 2005
"... We present several results on the complexity of various forms of Sperner’s Lemma. In the blackbox model of computing, we exhibit a deterministic algorithm for Sperner problems over pseudomanifolds of arbitrary dimension. The query complexity of our algorithm is essentially linear in the separation ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
(Show Context)
We present several results on the complexity of various forms of Sperner’s Lemma. In the blackbox model of computing, we exhibit a deterministic algorithm for Sperner problems over pseudomanifolds of arbitrary dimension. The query complexity of our algorithm is essentially linear in the separation number of the skeleton graph of the manifold and the size of its boundary. As a corollary we get an O ( √ n) deterministic query algorithm for the blackbox version of the problem 2DSPERNER, a well studied member of Papadimitriou’s complexity class PPAD. This upper bound matches the Ω ( √ n) deterministic lower bound of Crescenzi and Silvestri. In another blackbox result we prove for the same problem an Ω ( 4 √ n) lower bound for its probabilistic, and an Ω ( 8 √ n) lower bound for its quantum query complexity, showing that all these measures are polynomially related. Finally we explicit Sperner problems on a 2dimensional pseudomanifold and prove that they are complete respectively for the classes PPAD, PPADS and PPA. This is the first time that a 2dimensional Sperner problem is proved to be complete for any of the polynomial parity argument classes. 1
Spectral algorithms
, 2009
"... Summary. Spectral methods refer to the use of eigenvalues, eigenvectors, singular values and singular vectors. They are widely used in Engineering, Applied Mathematics and Statistics. More recently, spectral methods have found numerous applications in Computer Science to “discrete ” as well “continu ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
Summary. Spectral methods refer to the use of eigenvalues, eigenvectors, singular values and singular vectors. They are widely used in Engineering, Applied Mathematics and Statistics. More recently, spectral methods have found numerous applications in Computer Science to “discrete ” as well “continuous” problems. This book describes modern applications of spectral methods, and novel algorithms for estimating spectral parameters. In the first part of the book, we present applications of spectral methods to problems from a variety of topics including combinatorial optimization, learning and clustering. The second part of the book is motivated by efficiency considerations. A feature of many modern applications is the massive amount of input data. While sophisticated algorithms for matrix computations have been developed over a century, a more recent development is algorithms based on “sampling on the fly ” from massive matrices. Good estimates of singular values and low rank approximations of the whole matrix can be provably derived from a sample. Our
Higher eigenvalues of graphs
"... We present a general method for proving upper bounds on the eigenvalues of the graph Laplacian. In particular, we show that for any positive integer k, the k th smallest eigenvalue of the Laplacian on a boundeddegree planar graph is O(k/n). This bound is asymptotically tight for every k, as it is ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
(Show Context)
We present a general method for proving upper bounds on the eigenvalues of the graph Laplacian. In particular, we show that for any positive integer k, the k th smallest eigenvalue of the Laplacian on a boundeddegree planar graph is O(k/n). This bound is asymptotically tight for every k, as it is easily seen to be achieved for planar grids. We also extend this spectral result to graphs with bounded genus, graphs which forbid fixed minors, and other natural families. Previously, such spectral upper bounds were only known for k = 2, i.e. for the Fiedler value of these graphs. In addition, our result yields a new, combinatorial proof of the celebrated result of Korevaar in differential geometry.
Combinatorial and algebraic tools for optimal multilevel algorithms
, 2007
"... This dissertation presents combinatorial and algebraic tools that enable the design of the first linear work parallel iterative algorithm for solving linear systems involving Laplacian matrices of planar graphs. The major departure of this work from prior suboptimal and inherently sequential approac ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
This dissertation presents combinatorial and algebraic tools that enable the design of the first linear work parallel iterative algorithm for solving linear systems involving Laplacian matrices of planar graphs. The major departure of this work from prior suboptimal and inherently sequential approaches is centered around: (i) the partitioning of planar graphs into fixed size pieces that share small boundaries, by means of a local ”bottomup ” approach that improves the customary ”topdown ” approach of recursive bisection, (ii) the replacement of monolithic global preconditioners by graph approximations that are built as aggregates of miniature preconditioners. In addition, we present extensions to the theory and analysis of Steiner tree preconditioners. We construct more general Steiner graphs that lead to natural linear time solvers for classes of graphs that are known a priori to have certain structural properties. We also present a graphtheoretic approach to classical algebraic multigrid algorithms. We show that their design can be
Finding Sparse Cuts via Cheeger Inequalities for Higher Eigenvalues
"... Cheeger’s fundamental inequality states that any edgeweighted graph has a vertex subset S such that its expansion (a.k.a. conductance of S or the sparsity of the cut (S, ¯ S)) is bounded as follows: φ(S) def w(S, ¯ S) min{w(S), w ( ¯ S)} � √ 2λ2, where w is the total edge weight of a subset or ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Cheeger’s fundamental inequality states that any edgeweighted graph has a vertex subset S such that its expansion (a.k.a. conductance of S or the sparsity of the cut (S, ¯ S)) is bounded as follows: φ(S) def w(S, ¯ S) min{w(S), w ( ¯ S)} � √ 2λ2, where w is the total edge weight of a subset or a cut and λ2 is the second smallest eigenvalue of the normalized Laplacian of the graph. We study three natural generalizations of the sparsest cut in a graph: • a partition of the vertex set into k parts that minimizes the sparsity of the partition (defined as the ratio of the weight of edges between parts to the total weight of edges incident to the smallest k − 1 parts); • a collection of k disjoint subsets S1,..., Sk that minimize maxi∈[k] φ(Si); • a subset of size O(1/k) of the graph with minimum expansion. Our main results are extensions of Cheeger’s classical inequality to these problems via higher eigenvalues of the graph Laplacian. In particular, for the sparsest kpartition, we prove that the sparsity is at most 8 √ λk log k where λk is the k th smallest eigenvalue of the normalized Laplacian matrix. For the k sparse cuts problem we prove that there exist ck disjoint subsets S1,..., Sck, such that max i φ(Si) � C√λk log k where c, C are suitable absolute constants; this leads to a similar bound for the smallset expansion problem, namely for any k, there is a subset S whose weight is at most a O(1/k) fraction of the total weight and φ(S) � C √ λk log k. The latter two results are the best possible in terms of the eigenvalues up to constant factors. Our results are derived via simple and efficient algorithms, and can themselves be viewed as generalizations of Cheeger’s method.