Monographs in Computer Science
Brzozowski and Seger, Asynchronous Circuits
Selig, Geometrical Methods in Robotics
Nielson [editor], ML with Concurrency
Castillo, Gutierrez, and Hadi, Expert Systems and Probabilistic Network Models
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Downey and Fellows, Parameterized Complexity
R.G. Downey M.R. Fellows
Series Editors: David Gries Department of Computer Science Cornell University Upson Hall Ithaca, NY 14853-7501 USA
M.A. Fellows University of Victoria, Victoria Department of Computer Science Victoria, V8W 3P4 Canada
Fred B. Schneider Department of Computer Science Cornell University Upson Hall Ithaca, NY 14853-7501 USA
Library of Congress Cataloging-in-Publication Data Downey, R. G. (Rod G.)
Parameterized complexity I R.G. Downey, M.R. Fellows. p. cm. - (Monographs in computer science)
Includes index. ISBN 978-1-4612-6798-0 ISBN 978-1-4612-0515-9 (eBook) DOI 10.1007/978-1-4612-0515-9 1. Computational complexity. I. Fellows, M. R. 11. Title.
111. Series. QA267.7.D68 1997 511.3--dc21 97-22882
Printed on acid-free paper.
© 1999 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc. in 1999 Softcover reprint of the hardcover 1st edition 1999 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or here­ after developed is forbidden. The use of general descriptive names, Irade names, trademarks, ete., in Ihis publieation, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Aet, may aecordingly be used freely by anyone.
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987654321
Preface
The idea for this book was conceived over the second bottle of Villa Maria's Caber­ net Medot '89, at the dinner of the Australasian Combinatorics Conference held at Palmerston North, New Zealand in December 1990, where the authors first met and discovered they had a number of interests in common. Initially, we embarked on a small project to try to formulate reductions to address the apparent parame­ terized intractability of DOMINATING SET, and to introduce a structure in which to frame our answers. Having spent several months trying to get the definitions for the reductions right (they now seem so obvious), we turned to our tattered copies of Garey and Johnson's work [239]. We were stunned to find that virtually none of the classical reductions worked in the parameterized setting. We then wondered if we'd be able to find any interesting reductions.
Several years, many more bottles, so many papers, and reductions later it [3] seemed that we had unwittingly stumbled upon what we believe is a truly central and new area of complexity theory. It seemed to us that the material would be of great interest to people working in areas where exact algorithms for a small range of parameters are natural and useful (e.g., Molecular Biology, VLSI design). The tractability theory was rich with distinctive and powerful techniques. The intractability theory seemed to have a deep structure and techniques all of its own. We also felt that the subject was mathematically beautiful and sufficiently mature that we should gather the material for the present monograph. The project turned out to be bigger than we had expected.
We have tried to make the material as self-contained as possible, while keeping the book's length relatively finite. In some cases, such as tree automata, we felt that there was no adequate text and hence a lot of background material is included. Also, we have tried to make the book accessible to several distinct audiences, which
vi Preface
seemed to suggest rather more background material. We particularly tried to make the text accessible to graduate students. There are a large number of exercises included. They have varying levels of difficulty, and some are research results.
Anyway the book is as it is. We hope you enjoy the result.
Acknowledgements
The authors have had a lot of support for this venture. Downey was supported by a U.S.IN.Z. Cooperative Science grant INT-9020558, the ARO through the Mathematical Sciences Institute in Ithaca, New York via DAAL-03-C-0027, and Cornell University as a Visiting Scholar in 1992 and 1995. Downey especially acknowledges the generous support of the New Zealand Marsden Fund for Basic Science. Fellows acknowledges the generous, flexible, and reliable support of the National Science and Engineering Research Council of Canada.
Each of the authors have had exceptional support and encouragement from their home universities and departments. Victoria University of Wellington has provided Downey with numerous IGC grants and the special grant RGNT 779.
We would like to thank our various co-workers in the Parameterized World: Karl Abrahamson, Hans Bodlaender, Leizhen Cai, Liming Cai, Jianer Chen, Bruce Kapron, Neil Koblitz, Mike Langston, Venkatesh Raman, Ken Regan, and Igor Shparlinski. A special thanks to Mike's students Kevin Cattell, Marco Cesati, Mike Dinneen, Patricia Evans, Mike Hallett, Todd Wareham, and Xiu Yan Liu for providing stimulation in this area and especially to Todd Wareham and Mike Hallett who gave extensive help with the preparation of the Compendium. Also, special thanks to Downey's student Mike Doyle who provided very extensive and detailed corrections, and provided many of the diagrams.
Thanks to Edith Hodgen who struggled with the final preparation. We'd also like to thank the following who provided corrections to the vari­
ous drafts of the book: Ivan Murray, David Gresham, Alistair McGlinchy, Joseph Ganley, Richard Coles, Kristin Downey, and Geoffrey LaForte.
Thanks of course to Martin Gilchrist, Alan Abrams, and Fred Bartlett at Springer-Verlag.
viii Acknowledgements
Finally, we'd like to take this chance to sincerely thank our long suffering families who survived our frequent absences and put up with two aging but still relatively able surfers. We dedicate the book to them. Downey also dedicates it to the memory of his father, Reginald Andrew Downey 1926-1994.
Contents
List of Figures xiii
1 Computers, Complexity, and Iutractability from the Parametric Point of View 1 1.1 Introduction......................... 1 1.2 The Role of Computational Complexity in Modem Science 2 1.3 The Story of Dr. O. Continued ... . . . . . . . . . . . . 5 1.4 Reworking the Foundations of Computational Complexity . 6 1.5 A Dea1 with the Devil . . . . . . . 7 1.6 How Parameters Arise in Practice. 1.7 A Distinctive Positive Toolkit . . . 1.8 0 No? .............. . 1.9 The Barometer of Parametric Intractability 1.10 Structural Aspects of Parameterized Complexity 1.11 An Overview of Current Research Horizons ..
I Parameterized Tractability
8 11 13 14 18 19
21
2.2 The Advice View 26
3 Some Ad Hoc Methods: The Methods of Bounded Search Tree and Problem Kernel 29 3.1 The Method of Bounded Search Trees ........... 29
3.1.1 The Basic Method. . . . . . . . . . . . . . . . . . 29 3.1.2 Heuristic Improvements, Shrinking the Search Tree 35
3.2 The Method of Reduction to a Problem Kernel . . . . . . 39 3.2.1 The Basic Method. . . . . . . . . . . . . . . . . 39 3.2.2 Hereditary Properties and Leizhen Cai's Theorem 43
4 Optimization Problems, Approximation Schemes, and Their Relation with FPT 49 4.1 Optimization Problems ............. 49 4.2 How FPT and Optimization Problems Relate . . 50 4.3 The Classes MAXSNP, MINP+rr 1 (h), and FPT 54
5 The Advice View Revisited and LOGSPACE 61
6 Methods via Automata and Bounded Treewidth 67 6.1 Classical Automata Theory . . . . . . . . 67
6.1.1 Deterministic Finite Automata . . 67 6.1.2 Nondeterministic Finite Automata 69 6.1.3 Regular Languages . . . . . . . . 76 6.1.4 The Myhill-Nerode Theorem and the Method of Test
Sets .. . . . . . . . . . 80 6.1.5 Classical Tree Automata 89
6.2 Treewidth............ 99 6.3 Bodlaender's Theorem. . . . . . 105 6.4 Parse Trees for Graphs of Bounded Treewidth and an Analog
of the MyhiII-Nerode Theorem 113 6.5 Courcelle's Theorem. . . . . . . . . . . . . 130
6.5.1 The Basic Theorem . . . . . . . . . 130 6.5.2 Implementing Courcelle's Theorem 136
6.6 Seese's Theorem. . . 138 6.7 Notes on MSI Theory . . . . . . . . . . . . 140
7 Well-Quasi-Orderings and the Robertson-Seymour Theorems 143 7.1 Basic wQO Theory. . . . . . . . . . . . . . . . . . . . 143 7.2 Thomas' Lemma. . . . . . . . . . . . . . . . . . . . . 156
7.2.1 Thomas' Lemma Fails for Path Decompositions 164 7.3 The Graph Minor Theorem for Bounded Treewidth .. 169 7.4 Excluding a Forest. . . . . . . . . . . . . . . . . . . . 178 7.5 Connections with Automata Theory and Boundaried Graphs . 181 7.6 A Sketch of the Proof of the Graph Minor Theorem . . . .. 189
Contents xi
7.7 Immersions and the Nash-Williams Conjecture. . . . 192 7.8 Applications of the Obstruction Principle and wQo's . 194 7.9 Effectivizations of Obstruction-Based Methods. . . . 201
7.9.1 Effectivization by Self-Reduction. . . . . . . 202 7.9.2 Effectivization by Obstruction Set Computation 205
8 Miscellaneous Techniques 211 8.1 Depth-First Search .................... 211 8.2 Bounded-Width Subgraphs, the Plehn-Voigt Theorem, and
Induced Subgraphs 215 8.3 Hashing........................... 220
II Parameterized Intractability 225
9 Reductions 227
10 The Basic Class W[l] and an Analog of Cook's Theorem 235
11 Some Other W[1]-Hardness Results 255
12 The W-Hierarchy 283
14 Fixed Parameter Analogs of PSPACE and k-Move Games 331
15 Provable Intractability: The Class X P 341
III Structural and Other Results 351
16 Another Basis for the W -Hierarchy, the Tradeoff-Theorem, and Randomized Reductions 353
17 Relationships with Classical Complexity and Limited Nondeterminism 363 17.1 Classical Complexity ....... . . . . . . . . . . . . .. 363 17.2 Nondeterminism in P, LOGNP, and the Cai-Chen Model and
Other Models ......................... 369
18 The Monotone and Antimonotone Collapse Theorems: MONOTONE
W[2t + 1] = W[2t] and ANTIMONOTONE W[2t + 2] = W[2t + 1] 377
19 The Structure of Languages Under Parameterized Reducibilities 389 19.1 Some Tools 389 19.2 Results............................. 415
xii Contents
IV Appendix
A A Problem Compendium and Guide to W.Hierarchy Completeness, Hardness, and Classification; and Some Research Horizons Al In FPT ....... . A2 In FPT (Nonuniform) A3 In Randomized FPT . A4 W[l]-Complete ... A5 W[l]-Hard, in W[2] . A6 W[l]-Hard, in W[P] . A7 W[l]-Hard ..... . A8 W[2]-Complete... A9 W[2]-Hard, in W[P] . A.lO W[2]-Hard ..... . All W[t]-Complete .. . A.l2 W[t]-Hard, for All t, in W[P] . A13 W[t]-Hard, for All t A14 W[SAT]-Hard ........ . A15 W[P]-Complete ....... . A16 AW[*] = AW[l] = AW[t]-Complete A17 A W[SAT]-Complete A18 A W[SAT]-Hard . A19 A W[P]-Complete A20 A W[P]-Hard A21 X P-Complete
B Research Horizons B.l A Lineup of FPT Suspects ................... . B.2 A Lineup of Tough Customers ................. . B.3 Connections Between Classical and Parameterized Complexity. BA Classification Gaps ................. . B.5 Structural Issues and Analogs of Classical Results .. B.6 The FPTToolkit and Further Business with the Devil
References
Index
439
441 441 451 452 453 458 460 460 463 465 466 467 468 468 473 473 476 478 478 479 480 480
481 481 483 486 486 487 487
489
517
List of Figures
1.1 The Problem of Cleaning up the Data ........... 2 1.2 Classically Considered, Most Problems Yield a Seemingly
Unavoidable Combinatorial Explosion . . . . . . . . 6 1.3 Parameterized Complexity Attempts to Confine the
Combinatorial Explosion . 7 1.4 Lichenism ........... 13 1.5 Three Vertex Set Problems .. 14 1.6 Getting a Grip on Intractability 19
2.1 Examples of FPT Problems . 25
3.1 DOMINATING SET . . . . . . 31 3.2 The Cases of Termination for ALMOST(X) 33 3.3 Increasing the Number of Leaves . . 42
6.1 Transition Diagram for Example 6.2 68 6.2 Transition Diagram for Example 6.5 70 6.3 The Deterministic Automaton for Example 6.5 72 6.4 A Nondeterministic Automaton with A Moves . 73 6.5 The Automaton Corresponding to Example 6.9 74 6.6 Case (b) of Lemma 6.17 .......... 78 6.7 Construction of an Automaton Accepting L 79 6.8 The Resulting Automaton. . 85 6.9 The Minimization Algorithm 86 6.10 Automaton for Exercise 6 88
xiv List of Figures
6.11 An Example of a Tree. . . . . . . 89 6.12 Gluing Labeled Trees . . . . . . . 91 6.13 Computation of a Tree Evaluation 92 6.14 An Example of Productions and a Derivation Based on These
Productions . . . . . . . . . . . . . . . . . 94 6.15 Examples of 2- and 3-Trees ........ 100 6.16 Example of Tree Decomposition of Width 2 101 6.17 Figure for Exercise 2 . . . . . . . . . . . 104 6.18 Composition Operators and Their Effects 114 6.19 Parsing Graphs of Treewidth 2 . . . . . . 115 6.20 Normalizing the Tree Decomposition .. 117 6.21 Partial Parsing of the Normalized Decomposition 118 6.22 Replacement for the Myhill-Nerode Theorem . . 122 6.23 Graphs for BANDWIDTH. . . . . . . . . . . . . . 124 6.24 Test Set of Graphs for HAMILTONICITY for Treewidth 3 126 6.25 Constrained Layout for CUTWIDTH ........... 128 6.26 Test Sets for Atomic Formulae on Three Boundaried Graphs 137 6.27 Examples of Grids ........... 139
7.1 Kuratowski's Obstructions for Planarity 147 7.2 A Contraction and a Topological Embedding 150 7.3 Construction of the New Tree of Bags 159 7.4 The Coles Graph ......... . . . . . . 164 7.5 A Path Decomposition of G ......... 164 7.6 Embedding Pathwidth t Tree Obstructions, Tree-t, in Binary
Trees. . . . . . . . . . . . . . . . 185 7.7 Path Decomposition of Width 5. . 188 7.8 Some Graphs Without a P6 Minor 189 7.9 The Immersion Ordering . . . . 193 7.10 Obstructions for Interval Graphs 195 7.11 A 2-Fold Cover of K5 . . . . . . 197
8.1 An Example of a Depth-First Spanning Tree 8.2 A Circus Graph of Size 4 8.3 Graph for Exercise 2
10.1 Examples of Circuits 10.2 Gadget for RED/BLUE NONBLOCKER
11.1 Overview of INDEPENDENT SET ~~ PERFECT CODE
11.2 Example of INDEPENDENT SET ~:,. PERFECT CODE
12.1 Gadget for CNFSAT Reducing to DOMINATING SET
12.2 A Topmost Layer of Large Or Gates. ..... .
13.1 Gadgets for DEGREE 3 SUBGRAPH ANNIHILATOR
212 214 214
List of Figures xv
13.2 Gadgets for Exercise 1 ................... 328 13.3 Gadget for WEIGHTED PLANAR CIRCUIT SATISFIABILITY 329
14.1 Gadgets for SHORT GEOGRAPHY 338
15.1 Pebble Moves Where X = F(X_I, Xo, XI), X E (Q x 1) U f', dE {-I, 0, I}. Initially a Pebble Is on [qfb] and t = nk, i = 1 344
15.2 A Set of Rules, Where a, b, and c Are Nodes and P(a, c) Is a Condition . . . . . . . . . . . . 344
15.3 Rules for the Pebble Game . . . 346 15.4 More Rules for the Pebble Game
17.1 The Even Case of Theorem 17 .6( a) 17.2 The Algorithm CSAT(t, h)-SIMULATOR
19.1 The Arithmetical Hierarchy ..... 19.2 The Assignment of Priorities and the Outcomes 19.3 The Priority Tree for Theorem 19.23 ..... .
347
390 399 422
1 Computers, Complexity, and Intractability from the Parametric Point of View
1.1 Introduction
The subject of this book is perhaps best introduced through a somewhat whimsical story.
Imagine that a scientist, Dr. 0, has collected a number of data points that sup­ port a new theory of some sort and is nearly ready to publish. Dr. 0 observes that some pairs of these observations are in conflict, although most are not. The problem naturally arises of whether there is a way to determine a minimum set of observations in the data set that would "explain" all of the inconsistencies and that perhaps deserve further consideration.
Because some sophistication about computational matters is now called for in most sciences, Dr. 0 has a nodding familiarity with the subject of computational complexity and even owns a copy of the classic book by Garey and Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness [239]. After a bit of thought, Dr. 0 realizes that the problem can be modeled by "dots and lines." The data points are represented by dots, and a conflict between two data points is represented by a line between the two corresponding dots. The problem is then to come up with a minimum number of dots whose removal would eliminate all the conflicts along the lines of Figure 1.1. Consulting with the computer scientist who is a member of the interdisciplinary research group to which Dr. 0 belongs, Dr. 0 is informed that this is indeed a well-known problem that goes by the name of VERTEX COVER and is NP-complete. In fact, it is one of the six famous and basic NP-complete problems singled out for attention by Garey and Johnson.
2 1. The Parametric Point of View
FIGURE 1.1. The Problem of Cleaning up the Data
"But my data set is pretty good!" laments Dr. O. "There are probably only 40 or 50 troublesome data points, maybe only 20 .... "
Apart from the fact that the problem is NP-complete (and hard to approximate! [41]), all that classical computational complexity has to offer is that for every fixed value of k, there is a polynomial time algorithm to determine whether removing k points suffices to clean up the data set-namely, the trivial brute-force algorithm of trying all k subsets. Indeed, knowing that classically the problem is NP-complete would seem to suggest that the only approach would be exhaustive search.
The running time of this exhaustive search algorithm is roughly proportional to nk, where n is the size of the data set. If we assume that Dr. 0 has between 500 and 1000 data points, then this brute-force algorithm is impractical when k is more than about 5, since the number of steps would then exceed 1015, and few realistic computations involve that much effort.
1.2 The Role of Computational Complexity in Modem Science
Computational complexity is often regarded, even by computer scientists (in­ cluding the majority, who are not theoretical computer scientists and whose concerns are typically quite practical), as a mathematically forbidding, isolated, and even largely irrelevant subdiscipline. It is our view, however, that the subject of computational complexity has a profound and even central role in modem science.
The reason for this is that theories of computational complexity frame and focus the efforts of algorithm designers in certain directions (e.g., the quest for polynomial-time algorithms) and often discourage the investment of effort both on problems that are identified as "inherently intractable" and ones that are formally
1.2 The Role of Computational Complexity in Modem Science 3
regarded as asymptotically well-solved and therefore uninteresting. What efforts are put toward the "inherently intractable" for the most part could be classified as trying to establish…