Limits to Parallel Computation: P-Completeness Theory RAYMOND GREENLAW University of New Hampshire H. JAMES HOOVER University of Alberta WALTER L. RUZZO University of Washington New York Oxford OXFORD UNIVERSITY PRESS 1995 This book is dedicated to our families, who already know that life is inherently sequential. Preface This book is an introduction to the rapidly growing theory of P- completeness — the branch of complexity theory that focuses on identifying the “hardest” problems in the class P of problems solv- able in polynomial time. P-complete problems are of interest because they all appear to lack highly parallel solutions. That is, algorithm designers have failed to find NC algorithms, feasible highly parallel solutions that take time polynomial in the logarithm of the problem size while using only a polynomial number of processors, for them. Consequently, the promise of parallel computation, namely that ap- plying more processors to a problem can greatly speed its solution, appears to be broken by the entire class of P-complete problems. This state of affairs is succinctly expressed as the following question: Does P equal NC ? Organization of the Material The book is organized into two parts: an introduction to P- completeness theory, and a catalog of P-complete and open prob- lems. The first part of the book is a thorough introduction to the theory of P-completeness. We begin with an informal introduction. Then we discuss the major parallel models of computation, describe the classes NC and P, and present the notions of reducibility and com- pleteness. We subsequently introduce some fundamental P-complete problems, followed by evidence suggesting why NC does not equal P. Next, we discuss in detail the primary P-complete problem, that of evaluating a Boolean circuit. Following this we examine several se- quential paradigms and their parallel versions. We describe a model for classifying algorithms as inherently sequential. We finish Part I with some general conclusions about practical parallel computation. viii PREFACE Because of the broad range of topics in the rapidly growing field of parallel computation, we are unable to provide a detailed treatment of several related topics. For example, we are unable to discuss parallel algorithm design and development in detail. For important and broad topics like this, we provide the reader with some references to the available literature. The second part of the book provides a comprehensive catalog of P-complete problems, including several unpublished and new P- completeness results. For each problem we try to provide the essen- tial idea underlying its P-complete reduction, and as much informa- tion and additional references about related problems as possible. In addition to the P-complete problems catalog, we provide a list of open problems, a list of problems in the class CC , and a list of problems in the class RNC . Using the Book in a Course The book has been designed so that it will be suitable as a text for a one semester graduate course covering topics in parallel computation. The first part of the book provides introductory material and the problems can easily be converted into numerous exercises. The book would be ideal for use in a seminar course focusing on P-completeness theory. It can also be used as a supplementary text for a graduate level course in complexity theory. Several of the chapters in the book could be covered in a graduate level course or the book could be used as a reference for courses in the theory of computation. This work could also be used as a rich source of sample problems for a variety of different courses. For the motivated student or researcher interested in learning about P-completeness, the book can be used effectively for self study. Additional Problems and Corrections In producing this book over a period of many years we have tried to be as accurate and thorough as possible. Undoubtedly, there are still some errors and omissions in our lists of problems. In anticipation of possible future printings, we would like to correct these errors and in- corporate additional problems. We welcome suggestions, corrections, new problems, and further references. For each of the P-complete problems, we are also interested in references to papers that provide PREFACE ix the best known parallel algorithms for the problem. Please send general comments, corrections, new problems, bibli- ography entries, and/or copies of the relevant papers to Ray Greenlaw at [email protected] Corrigenda and additions to this work can by found in the directory pub/hoover/P-complete at ftp.cs.ualberta.ca and via the World Wide Web in Jim Hoover’s home page at http://www.cs.ualberta.ca/ Research Support The authors gratefully acknowledge financial support from the fol- lowing organizations: Ray Greenlaw’s research was supported in part by the National Science Foundation grant CCR-9209184. Jim Hoover’s research was supported by the Natural Sciences and Engineering Research Council of Canada grant OGP 38937. Larry Ruzzo’s research was supported in part by NSF grants ECS-8306622, CCR-8703196, CCR-9002891, and by NSF/DARPA grant CCR-8907960. A portion of this work was performed while visiting the University of Toronto, whose hospitality is also gratefully acknowledged. Acknowledgments An extra special thanks to Martin Tompa who helped keep this project alive when it was close to dying. Martin has provided us with many useful suggestions over the years — all of which have helped to make this a better book. A special thanks to Anne Condon and Alex Sch¨afferfor carefully reading drafts of the book. Their suggestions helped to improve specific sections. We wish to thank the following for contributing new problems, pointing us to appropriate literature, and providing us with cor- rections and suggestions on drafts of the manuscript: Richard An- derson, Jos´eBalc´azar,David Barrington, Paul Beame, Erik Brisson, Anne Condon, Stephen Cook, Derek Corneil, Pilar de la Torre, Larry Denenberg, Sergio De Agostino, John Ellis, Arvind Gupta, John x PREFACE Hershberger, David Johnson, Marek Karpinski, Tracy Kimbrel, Dex- ter Kozen, Luc Longpr´e,Mike Luby, Jon Machta, Andrew Malton, Pierre McKenzie, Satoru Miyano, Christos Papadimitriou, Teresa Przytycka, John Reif, Alex Sch¨affer,Roger Simons, Jack Snoeyink, Paul Spirakis, Iain Stewart, Larry Stockmeyer, Ashok Subramanian, Eva Tardos, Shang-Hua Teng, Martin Tompa, H. Venkateswaran, Joachim von zur Gathen, and Thomas Zeugmann. We want to thank the reviewers who provided us with general comments about the organization of the book as well as detailed comments. We have tried to incorporate as many of their suggestions as possible although not all of them were feasible for us. We want to thank Don Jackson at Oxford University Press. Don has been very helpful at each step in the production of this book and has been a great person to work with. Finally, we would like to thank Mike Garey and David Johnson whose book Computers and Intractability: A Guide to the Theory of NP-completeness [113] has served as a model for this work. November 1994 Ray Greenlaw Jim Hoover Larry Ruzzo PREFACE xi Authors’ Addresses Raymond Greenlaw Department of Computer Science University of New Hampshire Durham, NH 03824 e-mail address: [email protected] world wide web home page: http://www.cs.unh.edu/ H. James Hoover Department of Computing Science University of Alberta Edmonton, Alberta, Canada T6G 2H1 e-mail address: [email protected] world wide web home page: http://www.cs.ualberta.ca/ Walter L. Ruzzo Department of Computer Science and Engineering, FR-35 University of Washington Seattle, WA 98195 e-mail address: [email protected] world wide web home page: http://www.cs.washington.edu/ Contents Preface vii Part I: Background and Theory 1 1 Introduction 3 1.1 Bandersnatch Design 3 1.2 Informal Background 8 1.2.1 Feasible, Highly Parallel, and Inherently Sequential Problems 8 1.2.2 Why is P-Completeness Important? 11 1.3 Some History 13 1.4 Related Works 16 1.5 Overview of This Book 17 2 Parallel Models of Computation 19 2.1 Introduction 19 2.2 The PRAM Model 21 2.3 The Boolean Circuit Model 26 2.3.1 Uniform Circuit Families 30 2.4 Circuits and PRAMs 33 3 Complexity 38 3.1 Search and Decision Problems 38 3.2 Complexity Classes 41 3.2.1 P, NC , FP, and FNC 44 3.2.2 The Classes NC k and FNC k 45 3.2.3 Random NC 46 3.3 Reducibility 46 3.3.1 Many-one Reducibility 47 xiv CONTENTS 3.3.2 Turing Reducibility 49 3.4 Other NC Compatible Reducibilities 52 3.4.1 NC k Turing Reducibility 52 3.4.2 Logarithmic Space Reducibility 54 3.5 Completeness 54 4 Two Basic P -Complete Problems 57 4.1 The Generic P-Complete Problem 57 4.2 The Circuit Value Problem 59 5 Evidence That NC Does Not Equal P 61 5.1 Introduction 61 5.2 General Simulations Are Not Fast 62 5.3 Fast Simulations Are Not General 63 5.4 Natural Approaches Provably Fail 68 5.5 Summary 69 6 The Circuit Value Problem 71 6.1 The Circuit Value Problem Is P-Complete 71 6.2 Restricted Versions of Circuit Value 75 7 Greedy Algorithms 87 7.1 Lexicographic Greedy Algorithms 87 7.2 Generic Greedy Algorithms 90 8 P -Complete Algorithms 94 8.1 Introduction 94 8.2 Inherently Sequential Algorithms 96 8.3 Applications of the Model 98 9 Two Other Notions of P -Completeness 100 9.1 Strong P-Completeness 100 9.2 Strict P-Completeness 103 10 Approximating P -Complete Problems 108 10.1 Introduction 108 10.2 Approximating LFMIS Is Hard 109 10.3 Approximation Schemes 111 11 Closing Remarks 114 CONTENTS xv Part II: A Compendium of Problems