Algorithms and Combinatorics

Transcription

1 Algorithms and Combinatorics Volume 27 Editorial Board William J. Cook Ronald Graham Bernhard Korte LászlóLovász Avi Wigderson Günter M. Ziegler For further volumes:

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3 Stasys Jukna Boolean Function Complexity Advances and Frontiers 123

4 Stasys Jukna University of Frankfurt Institute of Informatics Robert-Mayer Str Frankfurt am Main Germany and Vilnius University Institute of Mathematics and Informatics Akademijos Vilnius Lithuania jukna@thi.informatik.uni-frankfurt.de ISSN ISBN e-isbn DOI / Springer Heidelberg Dordrecht London New York Library of Congress Control Number: Mathematics Subject Classification (2010): 68R05, 68Q17, 94C10 c Springer-Verlag Berlin Heidelberg 2012 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printed on acid-free paper Springer is part of Springer Science+Business Media (

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7 Preface Go to the roots of calculations! Group the operations. Classify them according to their complexities rather than their appearances! This, I believe, is the mission of future mathematicians. Evariste Galois Computational complexity theory is the study of the inherent hardness or easiness of computational tasks. Research in this theory has two main strands. One of these strands structural complexity deals with high-level complexity questions: is space a more powerful resource than time? Does randomness enhance the power of efficient computation? Is it easier to verify a proof than to construct one? So far we do not know the answers to any of these questions; thus most results in structural complexity are conditional results that rely on various unproven assumptions, like P NP. The second strand concrete complexity or circuit complexity deals with establishing lower bounds on the computational complexity of specific problems, like multiplication of matrices or detecting large cliques in graphs. This is essentially a low-level study of computation; it typically centers around particular models of computation such as decision trees, branching programs, boolean formulas, various classes of boolean circuits, communication protocols, proof systems and the like. This line of research aims to establish unconditional lower bounds, which rely on no unproven assumptions. This book is about the life on the second strand circuit complexity with a special focus on lower bounds. It gives self-contained proofs of a wide range of unconditional lower bounds for important models of computation, covering many of the gems of the field that have been discovered over the past several decades, right up to results from the last year or two. More than 20 years have passed since thewell-knownbookson circuitcomplexitybysavage (1976), Nigmatullin (1983), Wegener (1987), and Dunne (1988) as well as a famous survey paper of Boppana and Sipser (1990) were written. I feel it is time to summarize the new developments in circuit complexity during these two decades. vii

8 viii Preface The book is mainly devoted to mathematicians wishing to get an idea of what is actually going on in this one of the hardest, but also mathematically cleanest fields of computer science, to researchers in computer science wishing to refresh their knowledge about the state of art in circuit complexity, as well as to students wishing to try their luck in circuit complexity. I have highlighted some of the most important proof arguments for circuit lower bounds, without trying to be encyclopedic. To keep the length of the book within reasonable limits, I was forced to focus on classical circuit models results on their randomized or algebraic versions receive less attention here. Also, I often compromise the numerical tightness of results in favor of clarity of argument. My goal is to present the big picture of existing lower bound methods, in the hope that the reader will be motivated to find new ones. More than 40 open problems, marked as Research Problems, are mentioned along the way. Most of them are of a combinatorial or combinatorial-algebraic flavor and can be attacked by students with no background in computational complexity. The book is meant to be approachable for graduate students in mathematics and computer science, and is self-contained. The text assumes a certain mathematical maturity but no special knowledge in the theory of computing. For non-mathematicians, all necessary mathematical background is collected in the appendix of the book. As in combinatorics or in number theory, the models and problems in circuit complexity are usually quite easy to state and explain, even for the layperson. Most often, their solution requires a clever insight, rather than fancy mathematical tools. I am grateful to Miklos Ajtai, Marius Damarackas, Andrew Drucker, Anna Gál, Sergey Gashkov, Dmitry Gavinsky, Jonathan Katz, Michal Koucky, Matthias Krause, Andreas Krebs, Alexander Kulikov, Meena Mahajan, Igor Sergeev, Hans Ulrich Simon, György Turán, and Sundar Vishwanathan for comments and corrections on the draft versions of the book. Sergey Gashkov and Igor Sergeev also informed me about numerous results available only in Russian. I am especially thankful to Andrew Drucker, William Gasarch, Jonathan Katz, Massimo Lauria, Troy Lee, Matthew Smedberg, Ross Snider, Marcos Villagra, and Ryan Williams for proofreading parts of the book and giving very useful suggestions concerning the contents. Their help was crucial when putting the finishing touches to the manuscript. The strong commitment of Andrew Drucker in organizing these final touches and proofreading more than a half of the book by himself cannot be acknowledged well enough. All remaining errors are entirely my fault. My sincere thanks to Georg Schnitger for his support during my stay in Frankfurt. Finally, I would like to acknowledge the German Research Foundation (Deutsche Forschungsgemeinschaft) for giving an opportunity to finish the book while working within the grant SCHN 503/5-1. My deepest thanks to my wife, Daiva, and my daughter, Indrė, for their patience. Frankfurt am Main/Vilnius August 2011 Stasys Jukna

9 Contents Part I The Basics 1 Our Adversary: The Circuit Boolean Functions Circuits Branching Programs Almost All Functions are Complex Circuits Approximation Complexity The Circuit Hierarchy Theorem Switching Networks and Formulas Invariant Classes So Where are the Complex Functions? On Explicitness Explicit Lower Bounds A 3n Lower Bound for Circuits Graph Complexity Clique Complexity of Graphs Star Complexity of Graphs A Constant Factor Away From P NP? Exercises Analysis of Boolean Functions Boolean Functions as Polynomials Real Degree of Boolean Functions The Fourier Transform Boolean 0=1 Versus Fourier 1 Representation Approximating the Values 0 and Approximation by Low-Degree Polynomials Sign-Approximation Sensitivity and Influences Exercises ix

10 x Contents Part II Communication Complexity 3 Games on Relations Communication Protocols and Rectangles Protocols and Tiling Games and Circuit Depth Monotone Depth Exercises Games on 0-1 Matrices Deterministic Communication Nondeterministic Communication Greedy Bounds Fooling-Set Bounds P D NP \ co-np for Fixed-Partition Games Clique vs. Independent Set Game Communication and Rank The Log-Rank Conjecture Known Gaps Small Rank Implies Large Discrepancy Rank and Chromatic Number Communication with Restricted Advice P NP \ co-np for Best-Partition Games Randomized Communication Distributional Complexity Lower Bound for the Disjointness Function Unbounded Error Communication and Sign-Rank Private vs. Public Randomness Exercises Multi-Party Games The Number-in-Hand Model The Approximate Set Packing Problem Application: Streaming Algorithms The Number-on-Forehead Model The Discrepancy Bound Generalized Inner Product Matrix Multiplication Best-Partition k-party Communication Exercises Part III Circuit Complexity 6 Formulas Size Versus Depth A Quadratic Lower Bound for Universal Functions

11 Contents xi 6.3 The Effect of Random Restrictions A Cubic Lower Bound Nechiporuk s Theorem Lower Bounds for Symmetric Functions Formulas and Rectangles Khrapchenko s Theorem Complexity is not Convex Complexity is not Submodular The Drag-Along Principle Bounds Based on Graph Measures Lower Bounds via Graph Entropy Formula Size, Rank and Affine Dimension Affine Dimension and Formulas Projective Dimension and Branching Programs Exercises Monotone Formulas The Rank Argument Lower Bounds for Quadratic Functions A Super-Polynomial Size Lower Bound Rank of Disjointness Matrices A Lower Bound for Paley Functions A log 2 n Depth Lower Bound for Connectivity Reduction to the Fork Game Lower Bound for the Fork Game An n 1=6 Depth Lower Bound for Clique Function An n 1=2 Depth Lower Bound for Matching Exercises Span Programs The Model The Power of Span Programs Power of Monotone Span Programs Threshold Functions The Weakness of Monotone Span Programs Self-Avoiding Families Characterization of Span Program Size Monotone Span Programs and Secret Sharing Exercises Monotone Circuits Large Cliques are Hard to Detect Construction of the Approximated Circuit Bounding Errors of Approximation Very Large Cliques are Easy to Detect Properties of -Critical Graphs The Monotone Switching Lemma

12 xii Contents 9.4 The Lower-Bounds Criterion Explicit Lower Bounds Detecting Triangles Graphs of Polynomials Circuits with Real-Valued Gates Criterion for Graph Properties Clique-Like Problems What About Circuits with NOT Gates? Razborov s Method of Approximations Construction of Legitimate Lattices A Lower Bound for Perfect Matching Error-Probability on Accepted Inputs Error-Probability on Rejected Inputs Exercises The Mystery of Negations When are NOT Gates Useless? Slice Functions Negated Inputs as New Variables Markov s Theorem Formulas Require Exponentially More NOT Gates Fischer s Theorem How Many Negations are Enough to Prove P NP? Exercises Part IV Bounded Depth Circuits 11 Depth-3 Circuits Why is Depth 3 Interesting? An Easy Lower Bound for Parity The Method of Finite Limits A Lower Bound for Majority NP co-np for Depth-3 Circuits Graph Theoretic Lower Bounds Depth-2 Circuits and Ramsey Graphs Depth-3 Circuits and Signum Rank Depth-3 Circuits with Parity Gates Threshold Circuits General Threshold Circuits Threshold Circuits of Depth Two Threshold Circuits of Depth Three Exercises Large-Depth Circuits Håstad s Switching Lemma Razborov s Proof of the Switching Lemma

13 Contents xiii 12.3 Parity and Majority Are Not in AC Majority of AC 0 Circuits Parity is Even Hard to Approximate Constant-Depth Circuits and Average Sensitivity Circuits with Parity Gates Circuits with Modular Gates Circuits with Symmetric Gates Rigid Matrices Require Large Circuits Exercises Circuits with Arbitrary Gates Entropy and the Number of Wires Entropy and Depth-Two Circuits Matrix Product Is Hard in Depth Two Restricted Matrix Product Is Easy in Depth Three Larger-Depth Circuits Linear Circuits for Linear Operators Circuits with OR Gates: Rectifier Networks Circuits with OR and AND Gates Asymptotic Bounds Non-linear Circuits for Linear Operators Relation to Circuits of Logarithmic Depth Exercises Part V Branching Programs 14 Decision Trees Adversary Arguments P D NP \ co-np for Decision Tree Depth Certificates, Sensitivity and Block Sensitivity Block Sensitivity Versus Certificate Complexity Block Sensitivity Versus Depth Sensitivity and Degree of Polynomials Sensitivity and Subgraphs of the n-cube Evasive Boolean Functions Decision Trees for Search Problems Linear Decision Trees Element Distinctness and Turán s Theorem P NP \ co-np for Decision Tree Size Spectral Lower Bound Explicit Lower Bounds Exercises General Branching Programs Nechiporuk s Lower Bounds Lower Bounds for Symmetric Functions

14 xiv Contents 15.2 Branching Programs Over Large Domains Counting Versus Nondeterminism A Surprise: Barrington s Theorem Oblivious Branching Programs Exercises Bounded Replication Read-Once Programs: No Replications P NP \ co-np for Read-Once Programs Branching Programs Without Null-Paths Parity Branching Programs Linear Codes Require Large Replication Expanders Require Almost Maximal Replication Quadratic Functions of Expanders are Hard Exercises Bounded Time The Rectangle Lemma A Lower Bound for Code Functions Proof of the Rectangle Lemma Exercises Part VI Fragments of Proof Complexity 18 Resolution Resolution Refutation Proofs Resolution and Branching Programs Lower Bounds for Tree-Like Resolution Tree-Like Versus Regular Resolution Lower Bounds for General Resolution Size Versus Width Tseitin Formulas Expanders Force Large Width Matching Principles for Graphs Exercises Cutting Plane Proofs Cutting Planes as Proofs Cutting Planes and Resolution Lower Bounds for Tree-Like CP Proofs Lower Bound for the Matching CNF Lower Bounds for General CP Proofs The Clique-Coloring Polytope Chvátal Rank Rank Versus Depth of CP Proofs

15 Contents xv 19.7 Lower Bounds on Chvátal Rank The Maximum Independent Set Problem The Set-Covering Problem The Knapsack Problem An Upper Bounds on the Proof Size General CP Proofs Cannot be Balanced Size Versus Depth of CP Proofs Integrality Gaps Exercises Epilogue Pseudo-Random Generators Natural Proofs The Fusion Method Indirect Proofs Williams Lower Bound Kannan s Lower Bound A Mathematical Background A.1 Basics and Notation A.2 Graphs A.3 Linear Algebra A.4 Probability Theory References Index