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University Microfilms International 300 North Zeeb Road Ann Arbor, Michigan 48106 USA St John's Road INFORMATION TO USERS This material was produced from a microfilm copy of the original document. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependant upon the quality of the original submitted. The following explanation of techniques is provided to help you understand markings or patterns which may appear on this reproduction. 1. The sign or "target" for pages apparently lacking from the document photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting thru an image and duplicating adjacent pages to insure you complete continuity. 2. When an image on the film is obliterated with a large round black mark, it is an indication that the photographer suspected that the copy may have moved during exposure and thus cause a blurred image. You w ill find a good image of the page in the adjacent frame. 3. When a map, drawing or chart, etc., was part of the material being photographed the photographer followed a definite method in "sectioning" the material. It is customary to begin photoing at the upper left hand corner of a large sheet and to continue photoing from left to right in equal sections with a small overlap. If necessary, sectioning is continued again — beginning below the first row and continuing on until complete. 4. The majority of users indicate that the textual content is of greatest value, however, a somewhat higher quality reproduction could be made from "photographs" if essential to the understanding of the dissertation. Silver prints of "photographs" may be ordered at additional charge by writing the Order Department, giving the catalog number, title, author and specific pages you wish reproduced. 5. PLEASE NOTE: Some pages may have indistinct print. Filmed as received. University Microfilms International 300 North Zeeb Road Ann Arbor, Michigan 48106 USA St John's Road. Tyter s Green High Wycombe. Bucks. England HP10 6HR 77- 17,109 LEGGETT, Ernest Wil1iam, Jr., 1948- TOOLS AND TECHNIQUES FOR CLASSIFYING NP-HARD PROBLEMS. The Ohio State University, Ph.D., 1977 Computer Science Xerox University MicrofilmsAnn f Arbor, Michigan 46106 TOOLS AND TECHNIQUES FOR CLASSIFYING NP-HARD PROBLEMS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Ernest Villlam Leggett, Jr., B.S., M.S. * ★ * * * The Ohio State University 1977 Reading Committee: Approved By Prof. Daniel J. Moore / Prof. H. William Buttelmann Prof. Lee J. White Adviser Department of Computer and Information Science For Mary and Heather ACKNOWLEDGMENTS This dissertation would not have been possible without the advice and encouragement of my advisor, Dan Moore, for which I am grateful. I would also like to thank the other members of my reading committee, Lee White and Bill Buttelmann, for their guidance. I appreciate the support and understanding my co-workers at The Ohio State University Instruction and Research Computer Center and Its director, Prof. Roy F. Reeves. Finally, I thank my daughter, Heather, for her pleasant diversions during the writing of this dissertation. Most of all, I thank my wife, Mary, for her patience and love. In her opinion, the degree should be made out in "joint name," and I cannot disagree. ii TABLE OF CONTENTS Page ACKNOWLEDGMENTS .............................................. ii LIST OF F I G U R E S .............................................. iv LIST OF N O T A T I O N ............................................ v Chapter I INTRODUCTION AND BASIC DEFINITIONS ............... 1 II REDUCIBILITIES.................................... 28 III CLASSIFYING HARD PROBLEMS ........................ 58 IV INTUITION, PHILOSOPHY AND INSIGHTS ............... Ik V CONCLUSION AND OPEN P R O B L E M S ..................... 91 Appendix A ON RELATIVIZATIONS NPA AND DTIMEA (2lln) ......... 101 B PROBLEM DEFINITIONS .............................. 105 BIBLIOGRAPHY ................................................ Ill iii LIST OF FIGURES Figure Page 1 The Polynomial Hierarchy .......................... 17 2 Inclusions Among Polynomial Relations on PH .... 39 3 Constructing A <"p B^ From A <”p 51 4 Network Flow from Knapsack Packing ................ 68 iv LIST OF NOTATION Page Page A A 5 P , NP 21 46 *pe’< > “pe< i 6 21 51 6 np ,P 25 essentially in = t*■*= “ t 6 unary 62 7 25 preserves the =n< » “log<i best 66 9 NP-complete 29 mB. , MB- 71 13 k k 32 Bk* Bk ’ Bk* Bw 13 SG c-approximable 92 NP-hard 34 arbitrarily SG- 11 approximable 93 proper-A; 34 13 preserves ,np subproblems 94 35 13 ■m p-isomorphism 95 14 E- preserving 37 L-polynomially 14 approximable 97 np < » < 37 *c =c A-density 97 14 maximal relation 40 16 transitive 16 subrelation 40 "P 40 16 16 CHAPTER I INTRODUCTION AND BASIC DEFINITIONS Introduction In recent years, study in the area of computational complexity has drifted from the question "can this be computed" to the question "can this be computed efficiently?" There is one particular group of well over one hundred problems that we know can be solved, theoretically, usually by time-consuming combinatorial searches. This group of problems contains a great many of extremely practical significance that many people would like to be able to solve efficiently. We briefly describe several of them in Appendix B. It is not the purpose of this thesis to dwell extensively on details of these problems. The interested reader may consult references [AHU 74, Ka 72, Sa 74, SG 76] for further lists, discussions of, and references to this collection of problems. The practical importance of this class of problems should not be underesti­ mated. One can pick up almost any copy of the SIAM Journal on Computing, Journal of the ACM, network or graph theory journal, or journal on oper­ ations research and find new problems in this class or discussions of improved methods for attacking old ones. This class of problems includes ones from graph theory and its applications, network flows, integer programming, data base record clustering, computer and job shop scheduling, program code optimization, artificial Intelligence, 1 2 mathematical logic, formal languages, program structuring and verifica­ tion, geometry, number theory, and many other diverse areas. Technically, these problems have been termed "NP-hard," with an important subclass being the "NP-complete" problems. We will define these notions precisely later. The important point is that all of these problems from so many different areas of Interest appear to require an exponential amount of time to solve exactly. With our current tech­ nology and the sizes of the problems we wish to solve, obtaining exact solutions to the problems with current methods is too expensive, in most instances. Other problems, for which it seems fiscally sound to obtain exact solutions, have algorithms with running times bounded by a low- degree polynomial function of the size of the input question. Using a special type of machine model, a so-called "nondeterministic" one, which allows us to make guesses to decide what to do next in solving an instance of a problem, we can solve the "NP-complete" problems in similar low-degree polynomial time bounds, if we make the assumption that we always make the correct guess in executing a "nondeterministic" method. The question of whether normal algorithmic, or "deterministic," methods that are efficient (i.e., polynomially bounded) can be constructed from the "nondeterministic," or fortuitous-guessing, methods for solving the "NP-complete" problems is considered by many to be the outstanding open problem in the theory of computing today [e.g., see AHU 74, Chapter 10]. It is often referred to as the "is P equal to NP?" problem. The underlying purpose of the research presented here is to hopefully, make a contribution to the study of the nature of nondetermlnism 3 and the ultimate solution of the P - NP? and related problems. We want to better understand what common features the many, diverse NP-hard problems have and what the true nature of nondeterminism Is. We feel strongly that the effort to synthesize the essence of difficulty of those problems will be most successful if problems of similar difficulty are grouped as precisely as possible. One proposed vehicle for such a grouping is the Polynomial Hierarchy of Meyer and Stockmeyer [MS 72], which we will define later in this chapter. In this thesis, we locate several subclasses of "NP-hard" problems, particularly optimization problems, in that hierarchy, demonstrating several methods we devise. Our methods utilize query or oracle machines, machines which, from time to time in the course of their computations, request informa­ tion from an external agent, or oracle. The information from these oracular queries is presumed to be given immediately at a nominal cost, unlike that from the fickle Greek, oracle at Delphi. By putting certain restrictions on the kinds of query information such procedures may receive, so as to capture what we intuitively feel is the relationship between nondeterminism and query machine procedures, we can use those restricted procedures to more precisely classify interesting, hard problems. As a side benefit, we obtain several interesting results concerning those query procedures, or reduction relations, as they are often called. In the course of the study of these problems and reductions, we have gained some intuitive insights into the nature of the difficulties involved. These are discussed in Chapter IV of this thesis. The 4 nontechnically-oriented reader may get the basic impact of this work by skimming the definitions and theorems of Chapters I, XI, and III, reading the relatively non-technical Chapter IV, and skimming the con­ clusions and open problems in Chapter V. Basic Definitions Many of our basic definitions are similar to those in [AHU 74].
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