We study integrality gaps and approximability of two closely related problems on directed graphs. Given a set V of n nodes in an underlying asymmetric metric and two specified nodes s and t, both problems ask to find ...
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ISBN:
(纸本)9780898717013
We study integrality gaps and approximability of two closely related problems on directed graphs. Given a set V of n nodes in an underlying asymmetric metric and two specified nodes s and t, both problems ask to find an s-t path visiting all other nodes In the asymmetric traveling salesman path problem (ATSPP), the objective is to minimize the total cost of this path. In the directed latency problem, the objective is to minimize the sum of distances on this path from s to each node. Both of these problems are NP-hard The best known approximation algorithms for ATSPP had ratio O(log n) [7,9] until the very recent result that improves it to O(log n/ log log n) [3,9]. However, only a bound of O(root n) for the integrality gap of its lineal programming relaxation has been known. For directed latency, the best previously known approximation algorithm has a guarantee of O(n(1/2+epsilon)), for any constant epsilon > 0 [23] We present a new algorithm for the ATSPP problem that has approximation ratio of O(log n), but whose analysis also bounds the integrality gap of the standard LP relaxation of ATSPP by the same factor. This solves an open problem posed in [7]. We then pursue a deeper study of this LP and its variations and their use in approximating directed latency. Our second major result is an O(log n)approximation to the directed latency problem This also places an O(log n) bound on the integrality gap of a new LP relaxation of the latency problem that we introduce
We consider the problem of approximating a set P of n points in R-d by a j-dimensional subspace under the l(p) measure, in which we wish to minimize the sum or l(p) distances from each point or P to this subspace More...
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ISBN:
(纸本)9780898717013
We consider the problem of approximating a set P of n points in R-d by a j-dimensional subspace under the l(p) measure, in which we wish to minimize the sum or l(p) distances from each point or P to this subspace More generally, the F-q(l(p))-subspace approximation problem asks for a j-subspace that in the sum of gal powers of l(p)-distances to this subspace, up to a multiplicative factor of (1 + epsilon) We develop techniques for subspace approximation, regression. and matrix approximation that can be used to deal with massive data sets in high dimensional spaces In particular, we develop coresets and sketches, i e small space representations that approximate the input point set P with respect to the subspace approximation problem. Our results are. A dimensionality reduction method that can be applied to F-q(l(p))-clustering and shape fitting problems, such as those in [8, 15] The first strong coreset for F-1(l(2))-subspace approximation in high-dimensional spaces, i e. of size polynomial in the dimension of the space coreset approximates the distances to any j-subspace (not lust the optimal one) A (1 + epsilon)-approximation algorithm for the j-dimensional F-1(l(2))-subspace approximation problem with running time nd(J/epsilon)(O(1)) + (n + d)2(poly(J/epsilon)) A streaming algorithm that maintains a coreset for the F-1(l(2))-subspace approximation problem and uses a space of d(2 root log n/epsilon(2)) (weighted) points Streaming algorithms for the above problems with bounded precision in the turnstile model, i e, when coordinates coordinates appear in an arbitrary older and undergo multiple updates We show that bounded precision can lead to further improvements We extend results of [7] for approximate linear regression, distances to subspace approximation, and optimal rank-J approximation, to error measures other than the Frobenius norm
We study the following problem Given a planar graph G and a planar drawing (embedding) of a subgraph of G, can such a drawing be extended to a planar drawing of the entire graph G? This problem fits the paradigm of ex...
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ISBN:
(纸本)9780898717013
We study the following problem Given a planar graph G and a planar drawing (embedding) of a subgraph of G, can such a drawing be extended to a planar drawing of the entire graph G? This problem fits the paradigm of extending a partial solution to a complete one which has been studied before in many different settings. Unlike many cases, in which the presence of a partial solution in the input makes hard an otherwise easy problem, we show that the planarity question remains polynomial-time solvable Our algorithm is based on several combinatorial lemmata which show that the planarity of partially embedded graphs meets the "on-cas" behaviour obvious necessary conditions for planarity ale also sufficient These conditions are expressed in terms of the interplay between (a) rotation schemes and containment relationships between cycles and (b) the decomposition of a graph into its connected, biconnected, and triconnected components. This implies that no dynamic programming is needed for a decision algorithm and that the elements of the decomposition can be processed independently. Further, by equipping the components of the decomposition with suitable data structures and by carefully splitting the problem into simpler subproblems, we improve our algorithm to reach linear-time complexity Finally, we consider several generalizations of the problem, e g minimizing the number of edges of the Partial embedding that need to be rerouted to extend it, and fugue that they are NP-hard Also, we show how our algorithm can be applied to solve related Graph Drawing problems
In the kernel clustering problem we are given a (large) n x n symmetric positive semidefinite matrix A = (a(ij)) with Sigma(n)(i=1) Sigma(n)(j=1) a(ij) = 0 and a (small) k x k symmetric positive semidefinite matrix B ...
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ISBN:
(纸本)9780898717013
In the kernel clustering problem we are given a (large) n x n symmetric positive semidefinite matrix A = (a(ij)) with Sigma(n)(i=1) Sigma(n)(j=1) a(ij) = 0 and a (small) k x k symmetric positive semidefinite matrix B = (b(ij)) The goal is to find a partition (S(1), ..., S(k)} of {1, n} which maximizes Sigma(k)(i=1) Sigma(k)(j=1) (Sigma(p, q)is an element of S(1) x S(1) a(pq)) b(ij) We design a polynomial time approximation algorithm that achieves an approximation ratio of R(B)(2)/C(B) where R(B) and C(B) are geometric parameters that depend only on the matrix 13, defined as follows if b(ij) = < v(i), v(j)> is the Gram matrix representation of B lot some v(1), v(k) is an element of R(k) then R(B) is the minimum radius of a Euclidean ball containing the points {v(1), ..., v(k)} The parameter C(B) is defined as the maximum over all measurable partitions {A(1), ..., A(k)} of R(k-1) the quantity Sigma(k)(i=1) Sigma(k)(j=1) b(ij) < z(i), z(j)>, where for i is an element of{1, ..., k} the vector z, is an element of R(k-1) is the Gaussian moment of A(i), i e z, =,1/(2 pi)((k-1)/2) integral A ie-||x||(2)(2)/2dx We also show that for eve!): epsilon> 0, achieving an approximation guarantee of (1 - epsilon) R(B)(2)/C(B) is Unique Games hard
We give near-tight bounds for estimating the edit distance between two non-repetitive strings (Ulam distance) with constant approximation, in sub-linear time. For two strings of length d and at edit distance R, our al...
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ISBN:
(纸本)9780898717013
We give near-tight bounds for estimating the edit distance between two non-repetitive strings (Ulam distance) with constant approximation, in sub-linear time. For two strings of length d and at edit distance R, our algorithm runs in time (O) over tilde (d/R + root d) and outputs a constant approximation to R. We also prove a matching lower bound (up to logarithmic terms). Both upper and lower bounds arc improvements over previous results from, respectively, [Andoni-Indyk-Krauthgamer, SODA'09] and [Batu-Ergun-Kilian-Magen-Raskhodnikova-Rubinfeld-Sami, STOC'03].
This talk will describe the auction that Google uses for allocation and pricing of TV ads. The talk describes the actual system and puts it in proper theoretical context.
ISBN:
(纸本)9780898717013
This talk will describe the auction that Google uses for allocation and pricing of TV ads. The talk describes the actual system and puts it in proper theoretical context.
Workshops held in Miami, Florida, January 21, 2006. The annual Workshop on Algorithm Engineering and Experiments (ALENEX) provides a forum for the presentation of original research in all aspects of algorithm enginee...
ISBN:
(纸本)9780898716108
Workshops held in Miami, Florida, January 21, 2006. The annual Workshop on Algorithm Engineering and Experiments (ALENEX) provides a forum for the presentation of original research in all aspects of algorithm engineering, including the implementation and experimental evaluation of algorithms and data structures. The workshop was sponsored by SIAM, the Society for Industrial and appliedmathematics, and SIGACT, the ACM Special Interest Group on Algorithms and Computation Theory. The aim of ANALCO is to provide a forum for the presentation of original research in the analysis of algorithms and associated combinatorial structures.
The articles of this book are written by leading experts in partial differential equations and their applications, who present overviews here of recent advances in this broad area of mathematics. The formation of sho...
ISBN:
(纸本)9780821842119
The articles of this book are written by leading experts in partial differential equations and their applications, who present overviews here of recent advances in this broad area of mathematics. The formation of shocks in fluids, modern numerical computation of turbulence, the breaking of the Einstein equations in a vacuum, the dynamics of defects in crystals, effects due to entropy in hyperbolic conservation laws, the Navier–Stokes and other limits of the Boltzmann equation, occupancy times for Brownian motion in a two dimensional wedge, and new methods of analyzing and solving integrable systems are some of this volume's subjects. The reader will find an exposition of important advances without a lot of technicalities and with an emphasis on the basic ideas of this field.
It is the task of computational biology to help elucidate the unique characteristics of biological systems. This process has barely begun, and many researchers are testing computational tools that have been used succe...
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ISBN:
(纸本)9780821839645
It is the task of computational biology to help elucidate the unique characteristics of biological systems. This process has barely begun, and many researchers are testing computational tools that have been used successfully in other fields. Mathematical and statistical network modeling is an important step toward uncovering the organizational principles and dynamic behavior of biological networks. Undoubtedly, new mathematical tools will be needed, however, to meet this challenge. The workhorse of this effort at present comprises the standard tools from appliedmathematics, which have proven to be successful for many problems. But new areas of mathematics not traditionally considered applicable are contributing other powerful tools. This volume is intended to introduce this topic to a broad mathematical audience. The aim is to explain some of the biology and the computational and mathematical challenges we are facing. The different chapters provide examples of how these challenges are met, with particular emphasis on nontraditional mathematical approaches. The volume features a broad spectrum of networks across scales, ranging from biochemical networks within a single cell to epidemiological networks encompassing whole cities. Chapter topics include phylogenetics and gene finding using tools from statistics and algebraic geometry, biochemical network inference using tools from computational algebra, control-theoretic approaches to drug delivery using differential equations, and interaction-based modeling and discrete mathematicsapplied to problems in population dynamics and epidemiology.
Since their emergence in 1917, tomography and inverse problems remain active and important fields that combine pure and appliedmathematics and provide strong interplay between diverse mathematical problems and applic...
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ISBN:
(纸本)9780821839300
Since their emergence in 1917, tomography and inverse problems remain active and important fields that combine pure and appliedmathematics and provide strong interplay between diverse mathematical problems and applications. The applied side is best known for medical and scientific use, in particular, medical imaging, radiotherapy, and industrial non-destructive testing. Doctors use tomography to see the internal structure of the body or to find functional information, such as metabolic processes, noninvasively. Scientists discover defects in objects, the topography of the ocean floor, and geological information using X-rays, geophysical measurements, sonar, or other data. This volume, based on the lectures in the Short Course The Radon Transform and Applications to Inverse Problems at the American Mathematical Society meeting in Atlanta, GA, January 3–4, 2005, brings together articles on mathematical aspects of tomography and related inverse problems. The articles cover introductory material, theoretical problems, and practical issues in 3-D tomography, impedance imaging, local tomography, wavelet methods, regularization and approximate inverse, sampling, and emission tomography. All contributions are written for a general audience, and the authors have included references for further reading.
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