Clifford Stein
Total Page：16
File Type：pdf, Size：1020Kb
Load more
Recommended publications
-
A New Approach to the Minimum Cut Problem
A New Approach to the Minimum Cut Problem DAVID R. KARGER Massachusetts Institute of Technology, Cambridge, Massachusetts AND CLIFFORD STEIN Dartmouth College, Hanover, New Hampshire Abstract. This paper presents a new approach to finding minimum cuts in undirected graphs. The fundamental principle is simple: the edges in a graph’s minimum cut form an extremely small fraction of the graph’s edges. Using this idea, we give a randomized, strongly polynomial algorithm that finds the minimum cut in an arbitrarily weighted undirected graph with high probability. The algorithm runs in O(n2log3n) time, a significant improvement over the previous O˜(mn) time bounds based on maximum flows. It is simple and intuitive and uses no complex data structures. Our algorithm can be parallelized to run in 51# with n2 processors; this gives the first proof that the minimum cut problem can be solved in 51#. The algorithm does more than find a single minimum cut; it finds all of them. With minor modifications, our algorithm solves two other problems of interest. Our algorithm finds all cuts with value within a multiplicative factor of a of the minimum cut’s in expected O˜(n2a) time, or in 51# with n2a processors. The problem of finding a minimum multiway cut of a graph into r pieces is solved in expected O˜(n2(r21)) time, or in 51# with n2(r21) processors. The “trace” of the algorithm’s execution on these two problems forms a new compact data structure for representing all small cuts and all multiway cuts in a graph. This data structure can be efficiently transformed into the more standard cactus representation for minimum cuts. -
Approximation Algorithms for 2-Stage Stochastic Optimization Problems
Algorithms Column: Approximation Algorithms for 2-Stage Stochastic Optimization Problems Chaitanya Swamy∗ David B. Shmoys† February 5, 2006 1 Column editor’s note This issue’s column is written by guest columnists, David Shmoys and Chaitanya Swamy. I am delighted that they agreed to write this timely column on the topic related to stochastic optimization that has received much attention recently. Their column introduces the reader to several recent results and provides references for further readings. Samir Khuller 2 Introduction Uncertainty is a facet of many decision environments and might arise for various reasons, such as unpredictable information revealed in the future, or inherent fluctuations caused by noise. Stochastic optimization provides a means to handle uncertainty by modeling it by a probability distribution over possible realizations of the actual data, called scenarios. The field of stochastic optimization, or stochastic programming, has its roots in the work of Dantzig [4] and Beale [1] in the 1950s, and has since increasingly found application in a wide variety of areas, including transportation models, logistics, financial instruments, and network design. An important and widely-used model in stochastic programming is the 2- stage recourse model: first, given only distributional information about (some of) the data, one commits on initial (first-stage) actions. Then, once the actual data is realized according to the distribution, further recourse actions can be taken (in the second stage) to augment the earlier solution and satisfy the revealed requirements. The aim is to choose the initial actions so as to minimize the expected total cost incurred. The recourse actions typically entail making decisions in rapid reaction to the observed scenario, and are therefore more costly than decisions made ahead of time. -
Approximation Algorithms
In Introduction to Optimization, Decision Support and Search Methodologies, Burke and Kendall (Eds.), Kluwer, 2005. (Invited Survey.) Chapter 18 APPROXIMATION ALGORITHMS Carla P. Gomes Department of Computer Science Cornell University Ithaca, NY, USA Ryan Williams Computer Science Department Carnegie Mellon University Pittsburgh, PA, USA 18.1 INTRODUCTION Most interesting real-world optimization problems are very challenging from a computational point of view. In fact, quite often, finding an optimal or even a near-optimal solution to a large-scale optimization problem may require com- putational resources far beyond what is practically available. There is a sub- stantial body of literature exploring the computational properties of optimiza- tion problems by considering how the computational demands of a solution method grow with the size of the problem instance to be solved (see e.g. Aho et al., 1979). A key distinction is made between problems that require compu- tational resources that grow polynomially with problem size versus those for which the required resources grow exponentially. The former category of prob- lems are called efficiently solvable, whereas problems in the latter category are deemed intractable because the exponential growth in required computational resources renders all but the smallest instances of such problems unsolvable. It has been determined that a large class of common optimization problems are classified as NP-hard. It is widely believed—though not yet proven (Clay Mathematics Institute, 2003)—that NP-hard problems are intractable, which means that there does not exist an efficient algorithm (i.e. one that scales poly- nomially) that is guaranteed to find an optimal solution for such problems. -
David B. Shmoys
CURRICULUM VITAE David B. Shmoys 231 Rhodes Hall 312 Comstock Road Cornell University Ithaca, NY 14850 Ithaca, NY 14853 (607)-257-7404 (607)-255-9146 [email protected] Education Ph.D., Computer Science, University of California, Berkeley, June 1984. Thesis: Approximation Algorithms for Problems in Scheduling, Sequencing & Network Design Thesis Committee: Eugene L. Lawler (chair), Dorit S. Hochbaum, Richard M. Karp. B.S.E., Electrical Engineering and Computer Science, Princeton University, June 1981. Graduation with Highest Honors, Phi Beta Kappa, Tau Beta Pi. Thesis: Perfect Graphs & the Strong Perfect Graph Conjecture Research Advisors: H. W. Kuhn, K. Steiglitz and D. B. West Professional Employment Laibe/Acheson Professor of Business Management & Leadership Studies, Cornell, 2013–present. Tang Visiting Professor, Department of Ind. Eng. & Operations Research, Columbia, Spring 2019. Visiting Scientist, Simons Institute for the Theory of Computing, University of California, Berkeley, Spring 2018. Director, School of Operations Research & Information Engineering, Cornell University, 2013–2017. Professor of Operations Research & Information Engineering and of Computer Science, Cornell, 2007–present. Visiting Professor, Sloan School of Management, MIT, 2010–2011, Summer 2015. Visiting Research Scientist, New England Research & Development Center, Microsoft, 2010–2011. Professor of Operations Research & Industrial Engineering and of Computer Science, Cornell University, 1997–2006. Visiting Research Scientist, Computer Science Division, EECS, University of California, Berkeley, 1999–2000. Associate Professor of Operations Research & Industrial Engineering and of Computer Science, Cor- nell University, 1996–1997. Associate Professor of Operations Research & Industrial Engineering, Cornell University, 1992–1996. 1 Visiting Research Scientist, Department of Computer Science, Eotv¨ os¨ University, Fall 1993. Visiting Research Scientist, Department of Computer Science, Princeton University, Spring 1993. -
K-Medians, Facility Location, and the Chernoff-Wald Bound
86 K-Medians, Facility Location, and the Chernoff-Wald Bound Neal E. Young* Abstract minimizez d + k We study the general (non-metric) facility-location and weighted k-medians problems, as well as the fractional cost(x) < k facility-location and k-medians problems. We describe a dist(x) S d natural randomized rounding scheme and use it to derive approximation algorithms for all of these problems. subject to ~-~/x(f,c) = 1 (Vc) For facility location and weighted k-medians, the re- =(f,c) < x(f) (vf, c) spective algorithms are polynomial-time [HAk + d]- and [(1 + e)d; In(n + n/e)k]-approximation algorithms. These x(f,c) >_ 0 (vf, c) performance guarantees improve on the best previous per- x(f) E {0,1} (vf) formance guarantees, due respectively to Hochbaum (1982) and Lin and Vitter (1992). For fractional k-medians, the al- gorithm is a new, Lagrangian-relaxation, [(1 + e)d, (1 + e)k]- Figure 1: The facility-location IP. Above d, k, x(f) and approximation algorithm. It runs in O(kln(n/e)/e 2) linear- x(f, c) are variables, dist(x), the assignment cost of x, time iterations. is ~-~Ic x(f, c)disz(f, c), and cost(x), the facility cost of For fractional facilities-location (a generalization of frac- tional weighted set cover), the algorithm is a Lagrangian- relaxation, ~(1 + e)k]-approximation algorithm. It runs in O(nln(n)/e ~) linear-time iterations and is essentially the The weighted k-medians problem is the same same as an unpublished Lagrangian-relaxation algorithm as the facility location problem except for the following: due to Garg (1998). -
Introduction to Algorithms, 3Rd
Thomas H. Cormen Charles E. Leiserson Ronald L. Rivest Clifford Stein Introduction to Algorithms Third Edition The MIT Press Cambridge, Massachusetts London, England c 2009 Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form or by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher. For information about special quantity discounts, please email special [email protected]. This book was set in Times Roman and Mathtime Pro 2 by the authors. Printed and bound in the United States of America. Library of Congress Cataloging-in-Publication Data Introduction to algorithms / Thomas H. Cormen ...[etal.].—3rded. p. cm. Includes bibliographical references and index. ISBN 978-0-262-03384-8 (hardcover : alk. paper)—ISBN 978-0-262-53305-8 (pbk. : alk. paper) 1. Computer programming. 2. Computer algorithms. I. Cormen, Thomas H. QA76.6.I5858 2009 005.1—dc22 2009008593 10987654321 Index This index uses the following conventions. Numbers are alphabetized as if spelled out; for example, “2-3-4 tree” is indexed as if it were “two-three-four tree.” When an entry refers to a place other than the main text, the page number is followed by a tag: ex. for exercise, pr. for problem, fig. for figure, and n. for footnote. A tagged page number often indicates the first page of an exercise or problem, which is not necessarily the page on which the reference actually appears. ˛.n/, 574 (set difference), 1159 (golden ratio), 59, 108 pr. jj y (conjugate of the golden ratio), 59 (flow value), 710 .n/ (Euler’s phi function), 943 (length of a string), 986 .n/-approximation algorithm, 1106, 1123 (set cardinality), 1161 o-notation, 50–51, 64 O-notation, 45 fig., 47–48, 64 (Cartesian product), 1162 O0-notation, 62 pr. -
Exact and Approximate Algorithms for Some Combinatorial Problems
EXACT AND APPROXIMATE ALGORITHMS FOR SOME COMBINATORIAL PROBLEMS A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Soroush Hosseini Alamdari May 2018 c 2018 Soroush Hosseini Alamdari ALL RIGHTS RESERVED EXACT AND APPROXIMATE ALGORITHMS FOR SOME COMBINATORIAL PROBLEMS Soroush Hosseini Alamdari, Ph.D. Cornell University 2018 Three combinatorial problems are studied and efficient algorithms are pre- sented for each of them. The first problem is concerned with lot-sizing, the second one arises in exam-scheduling, and the third lies on the intersection of the k-median and k-center clustering problems. BIOGRAPHICAL SKETCH Soroush Hosseini Alamdari was born in Malayer, a town in Hamedan province of Iran right on the outskirts of Zagros mountains. After receiving a gold medal in the Iranian National Olympiad in Informatics he went to Sharif University of Technology in Tehran to study Computer Engineering. After completing his BSc he went to the University of Waterloo and received his MMath in computer science. Then after a short period of working in industry, he came back to com- plete his PhD in computer science here at the Cornell University. iii ACKNOWLEDGEMENTS First and foremost I would like to thank my Adviser Prof. David Shmoys who accepted me as a student, taught me so many fascinating things, and supported me in whatever endeavor I took on. Without his care and attention this docu- ment would simply not be possible. I would also like to thank all my friends, colleagues and co-authors. -
Approximation Algorithms for NP-Hard Optimization Problems
Approximation algorithms for NP-hard optimization problems Philip N. Klein Neal E. Young Department of Computer Science Department of Computer Science Brown University Dartmouth College Chapter 34, Algorithms and Theory of Computation Handbook c 2010 Chapman & Hall/CRC 1 Introduction In this chapter, we discuss approximation algorithms for optimization problems. An optimization problem consists in finding the best (cheapest, heaviest, etc.) element in a large set P, called the feasible region and usually specified implicitly, where the quality of elements of the set are evaluated using a func- tion f(x), the objective function, usually something fairly simple. The element that minimizes (or maximizes) this function is said to be an optimal solution of the objective function at this element is the optimal value. optimal value = minff(x) j x 2 Pg (1) A example of an optimization problem familiar to computer scientists is that of finding a minimum-cost spanning tree of a graph with edge costs. For this problem, the feasible region P, the set over which we optimize, consists of spanning trees; recall that a spanning tree is a set of edges that connect all the vertices but forms no cycles. The value f(T ) of the objective function applied to a spanning tree T is the sum of the costs of the edges in the spanning tree. The minimum-cost spanning tree problem is familiar to computer scientists because there are several good algorithms for solving it | procedures that, for a given graph, quickly determine the minimum-cost spanning tree. No matter what graph is provided as input, the time required for each of these algorithms is guaranteed to be no more than a slowly growing function of the number of vertices n and edges m (e.g. -
DISCRETE MATHEMATICS for COMPUTER SCIENTISTS This Page Intentionally Left Blank DISCRETE MATHEMATICS for COMPUTER SCIENTISTS
DISCRETE MATHEMATICS FOR COMPUTER SCIENTISTS This page intentionally left blank DISCRETE MATHEMATICS FOR COMPUTER SCIENTISTS Clifford Stein Columbia University Robert L. Drysdale Dartmouth College Kenneth Bogart Addison-Wesley Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto Delhi Mexico City Sao Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo Editor in Chief: Michael Hirsch Editorial Assistant: Stephanie Sellinger Director of Marketing: Margaret Whaples Marketing Coordinator: Kathryn Ferranti Managing Editor: Jeffrey Holcomb Production Project Manager: Heather McNally Senior Manufacturing Buyer: Carol Melville Media Manufacturing Buyer: Ginny Michaud Art Director: Linda Knowles Cover Designer: Elena Sidorova Cover Art: Veer Media Project Manager: Katelyn Boller Full-Service Project Management: Bruce Hobart, Laserwords Composition: Laserwords Credits and acknowledgments borrowed from other sources and reproduced, with permission, in this textbook appear on appropriate page within text. The programs and applications presented in this book have been included for their instructional value. They have been tested with care, but are not guaranteed for any particular purpose. The publisher does not offer any warranties or representations, nor does it accept any liabilities with respect to the programs or applications. Copyright © 2011. Pearson Education, Inc., publishing as Addison-Wesley, 501 Boylston Street, Suite 900, Boston, Massachusetts 02116. All rights reserved. Manufactured in the United States of America. This publication is protected by Copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. -
Thomas H. Cormen Current Position Research Interests Education Honors and Awards
Thomas H. Cormen Department of Computer Science 6211 Sudikoff Laboratory Dartmouth College Hanover, NH 03755-3510 (603) 646-2417 [email protected] http://www.cs.dartmouth.edu/thc/ Current Position Professor of Computer Science. Research Interests Algorithm engineering, parallel computing, speeding up computations with high latency, Gray codes. Education Massachusetts Institute of Technology, Cambridge, Massachusetts Ph.D. in Electrical Engineering and Computer Science, February 1993. Thesis: “Virtual Memory for Data-Parallel Computing.” Advisor: Charles E. Leiserson. Minor: Engineering management and entrepreneurship. S.M. in Electrical Engineering and Computer Science, May 1986. Thesis: “Concentrator Switches for Routing Messages in Parallel Computers.” Advisor: Charles E. Leiserson. Princeton University, Princeton, New Jersey B.S.E. summa cum laude in Electrical Engineering and Computer Science, June 1978. Honors and Awards ACM Distinguished Educator, 2009. McLane Family Fellow, Dartmouth College, 2004–2005. Jacobus Family Fellow, Dartmouth College, 1998–1999. Dartmouth College Class of 1962 Faculty Fellowship, 1995–1996. Adopted Member, Dartmouth College Class of 1962, 2015. Professional and Scholarly Publishing Award in Computer Science and Data Processing, Association of American Publishers, 1990. Distinguished Presentation Award, 1987 International Conference on Parallel Processing, St. Charles, Illinois. Best Presentation Award, 1986 International Conference on Parallel Processing, St. Charles, Illinois. National Science Foundation Fellowship. Elected to Phi Beta Kappa, Tau Beta Pi, Eta Kappa Nu. Prepared on March 28, 2020. 1 Professional Experience Dartmouth College, Hanover, New Hampshire Chair, Department of Computer Science, July 2009–July 2015. Professor, Department of Computer Science, July 2004–present. Director of the Dartmouth Institute for Writing and Rhetoric, July 2007–June 2008. -
Lecture Notes
Chapter 1 An introduction to approximation algorithms These lecture notes are for the course Advanced Algo- rithms (2016-2017) and are supplementary to the book `The design of approxiation algorithms' by David Shmoys and David Williamson. See the book for missing definitions. These notes only cover a part of the sections of the book. If you find an error/typo please let me know ([email protected]) and I will update de file. Contents 1 An introduction to approximation algorithms 3 1.1 Approximation Algorithms . .3 1.2 Introduction to techniques of LP . .3 1.3 A deterministic rounding algorithm . .6 1.4 Rounding a dual solution . .8 1.5 The primal-dual method . .9 1.6 A greedy algorithm . 11 1.7 A randomized rounding algorithm . 14 2 Greedy and Local search 17 2.1 Single machine scheduling . 17 2.2 The k-center problem. 18 2.3 Parallel machine scheduling. 19 2.4 The traveling salesman problem. 22 3 Rounding data and dynamic programming 26 3.1 The knapsack problem . 27 3.2 Scheduling on identical machines . 29 4 Deterministic rounding of linear programs 34 4.1 Sum of completion times on a single machine. 34 4.2 Weighted sum of completion times. 35 4.3 Using the ellipsoid method . 38 4.4 Prize-collecting Steiner Tree . 39 4.5 Uncapacitated facility location . 41 1 Chapter 1 An introduction to approximation algorithms 5 Random sampling and randomized rounding of LP's 44 5.1 Max Sat and Max Cut . 44 5.2 Derandomization. 47 5.3 Flipping biased coins . -
On the Complexity of Linear Programming
CHAPTER 6 On the complexity of linear programming Nimrod Megiddo Abstract: This is a partial survey of results on the complexity of the lin- ear programming problem since the ellipsoid method. The main topics are polynomial and strongly polynomial algorithms, probabilistic analy- sis of simplex algorithms, and recent interior point methods. 1 Introduction Our purpose here is to survey theoretical developments in linear program- ming, starting from the ellipsoid method, mainly from the viewpoint of computational comp1exity.l The survey does not attempt to be complete and naturally reflects the author's perspective, which may differ from the viewpoints of others. Linear programming is perhaps the most successful discipline of Oper- ations Research. The standard form of the linear programming problem is to maximize a linear function cTx (c,x E Rn)over all vectors x such that Ax = b and x r 0. We denote such a problem by (A,b, c).Currently, the main tool for solving the linear programming problem in practice is the class of simplex algorithms proposed and developed by Dantzig [43]. However, applications of nonlinear programming methods, inspired by Karmarkar's work [79], may also become practical tools for certain classes of linear programming problems. Complexity-based questions about linear programming and related parameters of polyhedra (see, e.g., [661) have been raised since the 1950s, before the field of computational complexity The author thanks Jetfrey Lagarias, David Shmoys, and Robert Wilson for helpful com- ments. ' The topic of linear programming has been interesting to economists not only due to its applicability to practical economic systems but also because of the many economic in- sights provided by the theory of linear programming.