DOCSLIB.ORG

Volume 20 Number 6 November 2011 Combinatorics, Probability & Computing Volume 20 Number 6 November 2011

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Volume 20 Number 6 November 2011 Combinatorics, Probability & Computing Volume 20 Number 6 November 2011 09635483_20-6.qxd 10/20/11 5:37 AM Page 1 Volume 20 Number 6 November 2011 & Computing Combinatorics, Probability Volume 20 Number 6 November 2011 CONTENTS Optimal Sequential Selection of a Unimodal Subsequence of a Random Sequence 799 ALESSANDRO ARLOTTO AND J. MICHAEL STEELE Editor-in-Chief: Béla Bollobás Computing the Partition Function for Perfect Matchings in a Hypergraph 815 Managing Editors: Paul Balister, Imre Leader, ALEXANDER BARVINOK AND ALEX SAMORODNITSKY VOLUME 20 NUMBER 6 November 2011 799–955 PAGES Editors: Oliver Riordan Turán Numbers of Multiple Paths and Equibipartite Forests 837 NEAL BUSHAW AND NATHAN KETTLE Noga Alon Graham Brightwell Counting Certain Pairings in Arbitrary Groups 855 Jennifer T. Chayes Y. O . HAMIDOUNE Alan Frieze Sums of Dilates in Groups of Prime Order 867 Zoltán Füredi ALAIN PLAGNE Timothy Gowers Analysis of Statistics for Generalized Stirling Permutations 875 Mark Jerrum MARKUS KUBA AND ALOIS PANHOLZER Jeff Kahn A Graph Integral Formulation of the Circuit Partition Polynomial 911 Yoshiharu Kohayakawa CRISTOPHER MOORE AND ALEXANDER RUSSELL Alexandr Kostochka Plünnecke’s Inequality 921 Greg Lawler GIORGIS PETRIDIS Tomasz Luczak The Final Size of the C4-Free Process 939 Colin McDiarmid MICHAEL E. PICOLLELLI Brendan McKay James Oxley Yuval Peres Alexander Razborov Vojteˇch Rödl Alex Scott Vera Sós Wojciech Szpankowski Andrew Thomason Dominic Welsh Peter Winkler Cambridge Journals Online For further information about this journal please go to the journal website at: journals.cambridge.org/cpc 0963-5483 Downloaded from https://www.cambridge.org/core. IP address: 170.106.33.22, on 25 Sep 2021 at 02:33:37, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S096354831100054X 09635483_20-6.qxd 10/20/11 5:37 AM Page 2 EDITOR-IN-CHIEF SUBSCRIPTIONS BÉLA BOLLOBÁS: Trinity College, Cambridge CB2 1TQ and Department of Mathematical Sciences, University of Combinatorics, Probability and Computing (ISSN: 0963-5483) is published bimonthly in 2011 in January, March, May, July, Memphis, Memphis, TN 38152, USA, Email: [email protected] and Email: [email protected] September and November by Cambridge University Press, The Edinburgh Building, Shaftesbury Road, Cambridge, CB2 8RU, MANAGING EDITORS UK / Cambridge University Press, 32 Avenue of the Americas, New York, N.Y. 10013–2473. The subscription price of Volume 20, 2011 PAUL BALISTER: Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA, is £380 net (USA, Canada and Mexico US$670) post free, institutions print and electronic; institutions electronic only £340 / $600; Email: [email protected] single parts cost £70 net (USA, Canada and Mexico US$123) plus postage. Prices include delivery by air. Six parts form a volume. Orders IMRE LEADER: Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, must be accompanied by payment, may be sent to any bookseller, subscription agent or to the publisher: Cambridge University Press, Wilberforce Road, Cambridge CB3 0WB, Email: [email protected] The Edinburgh Building, Shaftesbury Road, Cambridge, CB2 8RU, or in the USA, Canada and Mexico to Cambridge University Press, OLIVER RIORDAN: Mathematical Institute, 24-29 St Giles’, Oxford, OX1 3LB, Journals Fulfillment Department, 100 Brook Hill Drive, West Nyack, New York 10994-2133. EU subscribers outside the UK who are Email: [email protected] not registered for VAT should add VAT at their county’s rate. VAT registered subscribers should provide their VAT registration number. JOURNAL EMAIL ADDRESS: [email protected] Japanese prices for institutions are available from Kinokuniya Company Ltd., P.O. Box 55, Chitose, Tokyo156, Japan. Copies of the EDITORIAL BOARD Journal for subscribers in the USA, Canada and Mexico are sent by air to New York. Periodical postage paid at New York, N.Y. and NOGA ALON: Department of Mathematics, Tel Aviv University, Ramat Aviv, 69978 Tel Aviv, Israel, Email: [email protected] additional mailing offices. POSTMASTER Send address changes to Combinatorics, Probability and Computing, Cambridge University Probabilistic methods in combinatorics GRAHAM BRIGHTWELL: Department of Mathematics, London School of Economics, Houghton Street, London WC2A 2AE, Press, 100 Brook Hill Drive, West Nyack, N.Y. 10994-2133. Email: [email protected] Claims for missing issues can only be considered if made immediately upon receipt of the subsequent issue. Copies of back numbers Probabilistic methods and partially ordered sets are available from Cambridge University Press. JENNIFER T. CHAYES: Microsoft Research, One Microsoft Way, Redmond, Washington 98052, USA, Email: [email protected] Percolation, statistical physics methods COPYING This journal is registered with the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA. 01923, USA. ALAN FRIEZE: Department of Mathematics, Carnegie-Mellon University, Pittsburgh, PA 15213, Email: [email protected] Random graphs, algorithms Organisations in the USA who are also registered with the C.C.C. may therefore copy material (beyond the limits permitted by section ZOLTA´N FÜREDI: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA, Email: [email protected] 107 and 108 of US copyright law) subject to payment to C.C.C. of the per-copy fee of $16.00. This consent does not extend to multiple Set systems, extremal graph theory copying for promotional or commercial purposes. Code 0963-5483/11 $16.00. Organisations authorised by the Copyright Licensing TIMOTHY GOWERS: Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, Email: [email protected] Agency may also copy material subject to the usual conditions. ISI Tear Sheet Service, 3501 Market Street, Philadelphia, Pennsylvania Combinatorial number theory, probabilistic combinatorics 19104, USA, is authorised to supply single copies of separate articles for private use only. MARK JERRUM: School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, UK, Email: [email protected] For all other use, permission should be sought from Cambridge University Press. Randomised algorithms, complexity theory Cover design by Gabriella Bollobás based on a tessellation by Andreas Dress. JEFF KAHN: Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA, Email: [email protected] instructions for authors Extra-combinatorial methods, hypergraphs YOSHIHARU KOHAYAKAWA: Instituto de Matemática e Estatística, University of São Paulo, Rua do Matão 1010, 05508-900 São Paulo, Brazil, Email: [email protected] Editorial policy The journal welcomes submissions in any of the areas of Combinatorics, Probability, or Computer Science, not just Random combinatorial structures, extremal problems those dealing explicitly with relations between the three. Its scope covers combinatorics in a broad sense, including classical and ALEXANDR KOSTOCHKA: Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL 61801, USA, algebraic graph theory, probabilistic methods, random structures, combinatorial probability and limit theorems for random Email: [email protected] Graphs, hypergraphs combinatorial structures; and the theory of algorithms, including complexity theory, randomised algorithms, probabilistic analysis of GREG LAWLER: Department of Mathematics, University of Chicago, 5734 South University Avenue, Chicago, IL 60637, USA, Email: algorithms, computational learning theory and optimisation. [email protected] Submission of manuscripts Authors are strongly encouraged to submit papers electronically, ideally in the form of a pdf or postscript Probability theory TOMASZ L⁄ UCZAK: Institute of Mathematics, Adam Mickiewicz University, Matejki 48/49, PL-60 769 Poznán, Poland, file. Papers should be sent to the editorial office at [email protected], or can alternatively be submitted through any member of Email : [email protected] Probabilistic methods, random structures the Editorial Board. Papers may also be submitted in hard copy: three copies should be sent accompanied by the author's address, COLIN McDIARMID: Department of Statistics, University of Oxford, 1 South Parks Road, Oxford OX1 3TG, Email : [email protected] Combinatorics, probability and algorithms telephone, fax number, and email address. BRENDAN MCKAY: Computer Science Department, Australian National University, G.P.O. Box 4, Canberra, ACT 2601, Australia, Submission of a paper is taken to imply that it has not been previously published and that it is not being considered for publication Email: [email protected] Probability theory, enumeration, asymptotics JAMES OXLEY: Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803–4918, USA, Email: [email protected] elsewhere. Authors of articles published in the journal assign copyright to Cambridge University Press and will receive a copyright Matroid theory assignment form for signature upon acceptance of their paper. YUVAL PERES: Department of Statistics, 367 Evans Hall, University of California Berkeley, CA 94720-3860, USA, Email: [email protected] Probability on graphs, random walks On final acceptance of a paper, authors will be asked to send LaTeX source code, along with figure files and author-defined macros, ALEXANDER RAZBOROV: Department of Computer Science,
Load more

Recommended publications
  • FOCS 2005 Program SUNDAY October 23, 2005
    FOCS 2005 Program SUNDAY October 23, 2005 Talks in Grand Ballroom, 17th floor Session 1: 8:50am – 10:10am Chair: Eva´ Tardos 8:50 Agnostically Learning Halfspaces Adam Kalai, Adam Klivans, Yishay Mansour and Rocco Servedio 9:10 Noise stability of functions with low influences: invari- ance and optimality The 46th Annual IEEE Symposium on Elchanan Mossel, Ryan O’Donnell and Krzysztof Foundations of Computer Science Oleszkiewicz October 22-25, 2005 Omni William Penn Hotel, 9:30 Every decision tree has an influential variable Pittsburgh, PA Ryan O’Donnell, Michael Saks, Oded Schramm and Rocco Servedio Sponsored by the IEEE Computer Society Technical Committee on Mathematical Foundations of Computing 9:50 Lower Bounds for the Noisy Broadcast Problem In cooperation with ACM SIGACT Navin Goyal, Guy Kindler and Michael Saks Break 10:10am – 10:30am FOCS ’05 gratefully acknowledges financial support from Microsoft Research, Yahoo! Research, and the CMU Aladdin center Session 2: 10:30am – 12:10pm Chair: Satish Rao SATURDAY October 22, 2005 10:30 The Unique Games Conjecture, Integrality Gap for Cut Problems and Embeddability of Negative Type Metrics Tutorials held at CMU University Center into `1 [Best paper award] Reception at Omni William Penn Hotel, Monongahela Room, Subhash Khot and Nisheeth Vishnoi 17th floor 10:50 The Closest Substring problem with small distances Tutorial 1: 1:30pm – 3:30pm Daniel Marx (McConomy Auditorium) Chair: Irit Dinur 11:10 Fitting tree metrics: Hierarchical clustering and Phy- logeny Subhash Khot Nir Ailon and Moses Charikar On the Unique Games Conjecture 11:30 Metric Embeddings with Relaxed Guarantees Break 3:30pm – 4:00pm Ittai Abraham, Yair Bartal, T-H.
    [Show full text]
  • Property Testing on Boolean Functions Thesis Proposal
    Property testing on boolean functions Thesis proposal Eric Blais Computer Science Department Carnegie Mellon University Pittsburgh, PA [email protected] May 20, 2010 Abstract Property testing deals with the following general question: given query access to some combinatorial object, what properties of the ob- ject can one test with only a small number of queries? In this thesis, we will study property testing on boolean functions. Specifically, we will focus on two basic questions in the field: Can we characterize the set of properties on boolean functions that can be tested with a constant number of queries? And can we determine the exact number of queries needed to test some fundamental properties of boolean functions? 1 Contents 1 Introduction 3 2 Definitions and notation 4 2.1 Boolean functions . 4 2.2 Property testing . 5 3 Characterization of testable properties 5 3.1 Related work . 6 3.2 Testing function isomorphism . 7 3.3 Completed work on testing function isomorphism . 7 3.4 Proposed research . 8 4 Exact query complexity bounds 10 4.1 Related work . 10 4.2 Completed work on testing juntas . 11 4.3 Proposed research . 11 5 Other proposed research 12 5.1 Alternative models of property testing . 13 5.2 Application of property testing ideas to other domains . 14 6 Suggested timeline 14 2 1 Introduction This thesis is primarily concerned with the study of boolean functions. Boolean functions play a central role in many areas of computer science: complexity theory [44], machine learning [19], coding theory [43], cryptog- raphy [18], circuit design [34], data structures [30], and combinatorics [16].
    [Show full text]
  • 2021 Leroy P. Steele Prizes
    FROM THE AMS SECRETARY 2021 Leroy P. Steele Prizes The 2021 Leroy P. Steele Prizes were presented at the Annual Meeting of the AMS, held virtually January 6–9, 2021. Noga Alon and Joel Spencer received the Steele Prize for Mathematical Exposition. Murray Gerstenhaber was awarded the Prize for Seminal Contribution to Research. Spencer Bloch was honored with the Prize for Lifetime Achievement. Citation for Mathematical Biographical Sketch: Noga Alon Exposition: Noga Alon Noga Alon is a Professor of Mathematics at Princeton and Joel Spencer University and a Professor Emeritus of Mathematics and The 2021 Steele Prize for Math- Computer Science at Tel Aviv University, Israel. He received ematical Exposition is awarded his PhD in Mathematics at the Hebrew University of Jeru- to Noga Alon and Joel Spencer salem in 1983 and had visiting and part-time positions in for the book The Probabilistic various research institutes, including the Massachusetts Method, published by Wiley Institute of Technology, Harvard University, the Institute and Sons, Inc., in 1992. for Advanced Study in Princeton, IBM Almaden Research Now in its fourth edition, Center, Bell Laboratories, Bellcore, and Microsoft Research The Probabilistic Method is an (Redmond and Israel). He joined Tel Aviv University in invaluable toolbox for both 1985, served as the head of the School of Mathematical Noga Alon the beginner and the experi- Sciences in 1999–2001, and moved to Princeton in 2018. enced researcher in discrete He supervised more than twenty PhD students. He serves probability. It brings together on the editorial boards of more than a dozen international through one unifying perspec- technical journals and has given invited lectures in numer- tive a head-spinning variety of ous conferences, including plenary addresses in the 1996 results and methods, linked to European Congress of Mathematics and in the 2002 Inter- applications in graph theory, national Congress of Mathematicians.
    [Show full text]
  • CURRICULUM VITAE Rafail Ostrovsky
    last updated: December 26, 2020 CURRICULUM VITAE Rafail Ostrovsky Distinguished Professor of Computer Science and Mathematics, UCLA http://www.cs.ucla.edu/∼rafail/ mailing address: Contact information: UCLA Computer Science Department Phone: (310) 206-5283 475 ENGINEERING VI, E-mail: [email protected] Los Angeles, CA, 90095-1596 Research • Cryptography and Computer Security; Interests • Streaming Algorithms; Routing and Network Algorithms; • Search and Classification Problems on High-Dimensional Data. Education NSF Mathematical Sciences Postdoctoral Research Fellow Conducted at U.C. Berkeley 1992-95. Host: Prof. Manuel Blum. Ph.D. in Computer Science, Massachusetts Institute of Technology, 1989-92. • Thesis titled: \Software Protection and Simulation on Oblivious RAMs", Ph.D. advisor: Prof. Silvio Micali. Final version appeared in Journal of ACM, 1996. Practical applications of thesis work appeared in U.S. Patent No.5,123,045. • Minor: \Management and Technology", M.I.T. Sloan School of Management. M.S. in Computer Science, Boston University, 1985-87. B.A. Magna Cum Laude in Mathematics, State University of New York at Buffalo, 1980-84. Department of Mathematics Graduation Honors: With highest distinction. Personal • U.S. citizen, naturalized in Boston, MA, 1986. Data Appointments UCLA Computer Science Department (2003 { present): Distinguished Professor of Computer Science. Recruited in 2003 as a Full Professor with Tenure. UCLA School of Engineering (2003 { present): Director, Center for Information and Computation Security. (See http://www.cs.ucla.edu/security/.) UCLA Department of Mathematics (2006 { present): Distinguished Professor of Mathematics (by courtesy). 1 Appointments Bell Communications Research (Bellcore) (cont.) (1999 { 2003): Senior Research Scientist; (1995 { 1999): Research Scientist, Mathematics and Cryptography Research Group, Applied Research.
    [Show full text]
  • Derandomized Graph Products
    Derandomized Graph Products Noga Alon ∗ Uriel Feigey Avi Wigderson z David Zuckermanx February 22, 2002 Abstract Berman and Schnitger [10] gave a randomized reduction from approximating MAX- SNP problems [24] within constant factors arbitrarily close to 1 to approximating clique within a factor of n (for some ). This reduction was further studied by Blum [11], who gave it the name randomized graph products. We show that this reduction can be made deterministic (derandomized), using random walks on expander graphs [1]. The main technical contribution of this paper is in lower bounding the probability that all steps of a random walk stay within a specified set of vertices of a graph. (Previous work was mainly concerned with upper bounding this probability.) This lower bound extends also to the case that different sets of vertices are specified for different time steps of the walk. 1 Introduction We present lower bounds on the probability that all steps of a random walk stay within a specified set of vertices of a graph. We then apply these lower bounds to amplify unapprox- imability results about certain NP-hard optimization problems. Our work was motivated by the problem of approximating the size of the maximum clique in graphs. This motivat- ing problem, which serves also as an example of how our lower bounds can be applied, is described in section 1.1. ∗Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv Uni- versity, Tel Aviv, Israel and AT & T Bell Labs, Murray Hill, NJ, 07974, USA. Email: [email protected].
    [Show full text]
  • Discrete Mathematics: Methods and Challenges 121 Ous Interesting Applications
    ICM 2002 Vol. I 119–135 · · Discrete Mathematics: Methods and Challenges Noga Alon∗ Abstract Combinatorics is a fundamental mathematical discipline as well as an es- sential component of many mathematical areas, and its study has experienced an impressive growth in recent years. One of the main reasons for this growth is the tight connection between Discrete Mathematics and Theoretical Com- puter Science, and the rapid development of the latter. While in the past many of the basic combinatorial results were obtained mainly by ingenuity and detailed reasoning, the modern theory has grown out of this early stage, and often relies on deep, well developed tools. This is a survey of two of the main general techniques that played a crucial role in the development of mod- ern combinatorics; algebraic methods and probabilistic methods. Both will be illustrated by examples, focusing on the basic ideas and the connection to other areas. 2000 Mathematics Subject Classification: 05-02. Keywords and Phrases: Combinatorial nullstellensatz, Shannon capacity, Additive number theory, List coloring, The probabilistic method, Ramsey theory, Extremal graph theory. 1. Introduction The originators of the basic concepts of Discrete Mathematics, the mathemat- ics of finite structures, were the Hindus, who knew the formulas for the number of arXiv:math/0212390v1 [math.CO] 1 Dec 2002 permutations of a set of n elements, and for the number of subsets of cardinality k in a set of n elements, already in the sixth century. The beginning of Combinatorics as we know it today started with the work of Pascal and De Moivre in the 17th century, and continued in the 18th century with the seminal ideas of Euler in Graph Theory, with his work on partitions and their enumeration, and with his interest in latin squares.
    [Show full text]
  • Self-Sorting SSD: Producing Sorted Data Inside Active Ssds
    Foreword This volume contains the papers presentedat the Third Israel Symposium on the Theory of Computing and Systems (ISTCS), held in Tel Aviv, Israel, on January 4-6, 1995. Fifty five papers were submitted in response to the Call for Papers, and twenty seven of them were selected for presentation. The selection was based on originality, quality and relevance to the field. The program committee is pleased with the overall quality of the acceptedpapers and, furthermore, feels that many of the papers not used were also of fine quality. The papers included in this proceedings are preliminary reports on recent research and it is expected that most of these papers will appear in a more complete and polished form in scientific journals. The proceedings also contains one invited paper by Pave1Pevzner and Michael Waterman. The program committee thanks our invited speakers,Robert J. Aumann, Wolfgang Paul, Abe Peled, and Avi Wigderson, for contributing their time and knowledge. We also wish to thank all who submitted papers for consideration, as well as the many colleagues who contributed to the evaluation of the submitted papers. The latter include: Noga Alon Alon Itai Eric Schenk Hagit Attiya Roni Kay Baruch Schieber Yossi Azar Evsey Kosman Assaf Schuster Ayal Bar-David Ami Litman Nir Shavit Reuven Bar-Yehuda Johan Makowsky Richard Statman Shai Ben-David Yishay Mansour Ray Strong Allan Borodin Alain Mayer Eli Upfal Dorit Dor Mike Molloy Moshe Vardi Alon Efrat Yoram Moses Orli Waarts Jack Feldman Dalit Naor Ed Wimmers Nissim Francez Seffl Naor Shmuel Zaks Nita Goyal Noam Nisan Uri Zwick Vassos Hadzilacos Yuri Rabinovich Johan Hastad Giinter Rote The work of the program committee has been facilitated thanks to software developed by Rob Schapire and FOCS/STOC program chairs.
    [Show full text]
  • Problems and Results in Extremal Combinatorics, Part I
    Problems and results in Extremal Combinatorics, Part I Noga Alon ∗ Abstract Extremal Combinatorics is an area in Discrete Mathematics that has developed spectacularly during the last decades. This paper contains a collection of problems and results in the area, including solutions or partial solutions to open problems suggested by various researchers in Extremal Graph Theory, Extremal Finite Set Theory and Combinatorial Geometry. This is not meant to be a comprehensive survey of the area, it is merely a collection of various extremal problems, which are hopefully interesting. The choice of the problems is inevitably somewhat biased, and as the title of the paper suggests I hope to write a related paper in the future. Each section of this paper is essentially self contained, and can be read separately. 1 Introduction Extremal Combinatorics deals with the problem of determining or estimating the maximum or min- imum possible cardinality of a collection of finite objects that satisfies certain requirements. Such problems are often related to other areas including Computer Science, Information Theory, Number Theory and Geometry. This branch of Combinatorics has developed spectacularly over the last few decades, see, e.g., [10], [29], and their many references. This paper contains a collection of problems and results in the area, including solutions or partial solutions to open problems suggested by various researchers in Extremal Graph Theory, Extremal Finite Set Theory and Combinatorial Geometry. This is not meant to be a comprehensive survey of the area, but rather a collection of several extremal problems, which are hopefully interesting. The techniques used include combinatorial, probabilistic and algebraic tools.
    [Show full text]
  • Arxiv:2012.06421V2 [Cs.LG] 21 Jul 2021
    When is Memorization of Irrelevant Training Data Necessary for High-Accuracy Learning? Gavin Brown∗ Mark Bun∗ Vitaly Feldman† Adam Smith∗ Kunal Talwar† July 23, 2021 Abstract Modern machine learning models are complex and frequently encode surprising amounts of information about individual inputs. In extreme cases, complex models appear to memorize entire input examples, including seemingly irrelevant information (social secu- rity numbers from text, for example). In this paper, we aim to understand whether this sort of memorization is necessary for accurate learning. We describe natural prediction problems in which every sufficiently accurate training algorithm must encode, in the pre- diction model, essentially all the information about a large subset of its training examples. This remains true even when the examples are high-dimensional and have entropy much higher than the sample size, and even when most of that information is ultimately irrel- evant to the task at hand. Further, our results do not depend on the training algorithm or the class of models used for learning. Our problems are simple and fairly natural variants of the next-symbol prediction and the cluster labeling tasks. These tasks can be seen as abstractions of text- and image- related prediction problems. To establish our results, we reduce from a family of one-way communication problems for which we prove new information complexity lower bounds. Additionally, we present synthetic-data experiments demonstrating successful attacks on logistic regression and neural network classifiers. arXiv:2012.06421v2 [cs.LG] 21 Jul 2021 ∗Computer Science Department, Boston University. {grbrown,mbun,ads22}@bu.edu. GB and AS are supported in part by NSF award CCF-1763786 as well as a Sloan Foundation research award.
    [Show full text]
  • The Shannon Capacity of a Union
    The Shannon Capacity of a union Noga Alon ∗ To the memory of Paul Erd}os Abstract For an undirected graph G = (V; E), let Gn denote the graph whose vertex set is V n in which two distinct vertices (u1; u2; : : : ; un) and (v1; v2; : : : ; vn) are adjacent iff for all i between 1 and n 1=n n either ui = vi or uivi E. The Shannon capacity c(G) of G is the limit limn (α(G )) , 2 !1 where α(Gn) is the maximum size of an independent set of vertices in Gn. We show that there are graphs G and H such that the Shannon capacity of their disjoint union is (much) bigger than the sum of their capacities. This disproves a conjecture of Shannon raised in 1956. 1 Introduction For an undirected graph G = (V; E), let Gn denote the graph whose vertex set is V n in which two distinct vertices (u1; u2; : : : ; un) and (v1; v2; : : : ; vn) are adjacent iff for all i between 1 and n either n 1=n n ui = vi or uivi E. The Shannon capacity c(G) of G is the limit limn (α(G )) , where α(G ) is 2 !1 the maximum size of an independent set of vertices in Gn. This limit exists, by super-multiplicativity, and it is always at least α(G). (It is worth noting that it is sometimes customary to call log c(G) the Shannon capacity of G, but we prefer to use here the above definition, following Lov´asz[12].) The study of this parameter was introduced by Shannon in [14], motivated by a question in Information Theory.
    [Show full text]
  • Large Networks and Graph Limits
    American Mathematical Society Colloquium Publications Volume 60 Large Networks and Graph Limits László Lovász Large Networks and Graph Limits http://dx.doi.org/10.1090/coll/060 American Mathematical Society Colloquium Publications Volume 60 Large Networks and Graph Limits László Lovász American Mathematical Society Providence, Rhode Island Editorial Board Lawrence C. Evans Yuri Manin Peter Sarnak (Chair) 2010 Mathematics Subject Classification. Primary 58J35, 58D17, 58B25, 19L64, 81R60, 19K56, 22E67, 32L25, 46L80, 17B69. For additional information and updates on this book, visit www.ams.org/bookpages/coll-60 ISBN-13: 978-0-8218-9085-1 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2012 by the author. All rights reserved. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 171615141312 To Kati as all my books Contents Preface xi Part 1.
    [Show full text]
  • Finding a Large Hidden Clique in a Random Graph ∗
    Finding a large hidden clique in a random graph ∗ Noga Alon y Michael Krivelevich z Benny Sudakov x Abstract We consider the following probabilistic model of a graph on n labeled vertices. First choose a random graph G(n; 1=2) and then choose randomly a subset Q of vertices of size k and force it to be a clique by joining every pair of vertices of Q by an edge. The problem is to give a polynomial time algorithm for finding this hidden clique almost surely for various values of k. This question was posed independently, in various variants, by Jerrum and by Kuˇcera. In this paper we present an efficient algorithm for all k > cn0:5, for any fixed c > 0, thus improving the trivial case k > cn0:5(log n)0:5. The algorithm is based on the spectral properties of the graph. 1 Introduction A clique in a graph G is a set of vertices any two of which are connected by an edge. Let w(G) denote the maximum number of vertices in a clique of G. The problem of determining or estimating w(G) and that of finding a clique of maximum size in G are fundamental problems in Theoretical Computer Science. The problem of computing w(G) is well known to be NP-hard [16]. The best known approximation algorithm for this quantity, designed by Boppana and Halld´orsson[8], has a performance guarantee of O(n=(log n)2), where n is the number of vertices in the graph. When the graph contains a large clique, there are better algorithms, and the best one, given in [3], shows that if w(G) exceeds n=k +m, where k is a fixed integer and m > 0, then one can find a clique of size Ω(~ m3=(k+1)) in polynomial time, where here the notation g(n) = Ω(~ f(n)) means, as usual, that g(n) Ω(f(n)=(log n)c) for some constant c independent of n.
    [Show full text]