Mayr E.W., Pr?mel H.J., Steger A. (eds.) Lectures on Proof Verification and Approximation Algorithms
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Lecture Notes in Computer Science
Edited by G. Goos, J. Hartmanis and J. van
Leeuwen
1367
Ernst W. Mayr Hans Jtirgen Pr6mel
Angelika Steger (Eds.)
Lectures on
Proof Verification
and Approximation
Algorithms
~ Springer
Series Editors
Gerhard Goos, Karlsruhe Universitg Germany
Juris Hartmanis, Cornell Universitg NY, USA
Jan van Leeuwen, Utrecht Universitg The Netherlands
Volume Editors
Ernst W. Mayr
Angelika Steger
Institut ftir Informatik, Technische Oniversitiit Mtinchen
D-80290 Miinchen, Germany
E-mail: { mayr, steger } @in formatik.tu-muenchen.de
Hans Jiirgen Pr/Smel
Institut ftir Informatik, Humboldt-Universitiit zu Berlin
D-10099 Berlin, Germany
E-mail: proemel@in formatik.hu-berlin.de
Cataloging-in-Publication data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Lectures
on proof verification and approximation algorithms /
Ernst W. May ... (ed.). - Berlin ; Heidelberg ; New York ; Barcelona ;
Budapest
; Hong Kong ; London ; Milan ; Paris ;
Santa Clara
;
Singapore ; Tokyo : Springer, 1998
(Lecture notes in computer science ; 1367)
ISBN 3-540-64201-3
CR Subject Classification (1991): E2, F.I.3, D.2.4, G.I.2, G.1.6, G.3, 1.3.5
ISSN 0302-9743
ISBN 3-540-64201-3 Springer-Verlag Berlin Heidelberg New York
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Preface
Proof Verification and Approximation Algorithms -
Hardly any area in theoret-
ical computer science has been more lively and flourishing during the last few
years. Different lines of research which had been developed independently of
each other over the years culminated in a new and unexpected characterization
of the well-known complexity class Af:P, based on probabilistically checking cer-
tain kinds of proofs. This characterization not only sheds new light on the class
AfT~ itself, it also allows proof of non-approximability results for optimization
problems which, for a long time, had seemed to be out of reach. This connection,
in turn, has motivated scientists to take a new look at approximating AlP-hard
problems as well - with quite surprising success. And apparently, these exciting
developments are far from being finished.
We therefore judged "Proof Verification and Approximation Algorithms" an
ideal topic for the first in a new series of research seminars for young scientists,
to be held at the International Conference and Research Center for Computer
Science at Schloi] Dagstuhl in Germany. This new series of seminars was estab-
lished by the German Society for Computer Science (Gesellschaft ffir Informatik,
GI) with the aim of introducing students and young scientists to important new
research areas and results not yet accessible in text books or covered in the
literature in a comprehensive way.
When we announced our seminar we encountered considerable interest and re-
ceived numerous responses. We were able to select 21 qualified doctoral students
and postdocs. Each participant then was requested to give a lecture, usually
based on several research articles or technical reports, and to submit, in prelim-
inary form and before the workshop began, an exposition of the topic assigned
to him/her. The actual workshop then took place April 21-25, 1997 at SchloB
Dagstuhl. All participants were very well prepared and highly motivated. We
heard excellent talks and had many interesting and stimulating discussions, in
the regular sessions as well as over coffee or some enlightening glass of wine after
dinner.
This volume contains revised versions of the papers submitted by the partici-
pants. The process of revision involved, among other things, unifying notation,
removing overlapping parts, adding missing links, and even combining some of
the papers into single chapters. The resulting text should now be a coherent
vl
and essentially self-contained presentation of the enormous recent progress facil-
itated by the interplay between the theory of probabilistically checkable proofs
and approximation algorithms. While it is certainly not a textbook in the usual
sense, we nevertheless believe that it can be helpful for all those who are just
starting out to learn about these subjects, and hopefully even to those looking
for a coherent treatment of the subject for teaching purposes.
Our workshop was sponsored generously, by Special Interest Group 0 (Fachbe-
reich "Grundlagen der Informatik") of the German Society for Computer Science
(GI) and by the International Conference and Research Center for Computer
Science (Internationales Begegnungs- und Forschungszentrum ffir Informatik,
IBFI) at SchloB Dagstuhl. We owe them and the staff at SchloB Dagstuhl many
thanks for a very successful and enjoyable meeting.
M/inchen, Berlin
September 1997
Ernst W. Mayr
Hans J/irgen Pr5mel
Angelika Steger
Prologue
Exam time. Assume you are the teaching assistant for some basic course with s
students, s very large. The setup for the exam is as follows:
(1) The exam consists of q yes/no questions.
(2) A student passes if
and only if
he or she answers all questions correctly.
You assume that, on average, you'll need at least half a second to check the
correctness of each answer. Since you expect the number of students to be close
to one thousand (it is a very popular basic course!) and since the number of
questions will be several hundred, a rough estimate shows that you are going to
spend almost a whole week grading the exam. Ooff.
Is there a faster way?
Certainly not in general: in the worst case you really might have to look at all
s. q answers in order to rule out a false decision. But what if we relax the second
condition slightly and replace it by
(2') A student definitely passes the exam if he or she answers all questions
correctly. A student who does not answer all questions correctly may pass
only with a small probability, say ~ 10 -a, independently of the answers he
or she gives.
Now you suddenly realize that the grading can actually be done in about 45s
seconds, even regardless of the actual number q of questions asked in the exam.
That is, a single day should suffice. Not too bad.
How is this possible? Find out by reading this book! And enjoy!
Table of Contents
Introduction .....................................................
1
1.
Introduction to the Theory of Complexity and
Approximation
Algorithms ..................................................
Thomas Janse~l
1.1 Introduction .............................................. 5
1.2 Basic Definitions .......................................... 6
1.3 Complexity Classes for Randomized Algorithms ............... 12
1.4 Optimization and Approximation ............................ 16
. Introduction to Randomized Algorithms ..................... 29
Artur Andrzejak
2.1 Introduction .............................................. 29
2.2 Las Vegas and Monte Carlo Algorithms ...................... 30
2.3 Examples of Randomized Algorithms ........................ 32
2.4 Randomized Rounding of Linear Programs ................... 34
2.5 A Randomized Algorithm for MAXSAT with Expected
Performance Ratio 4/3 ..................................... 37
.
Derandomization
............................................ 41
Detlef Sieling
3.1 Introduction .............................................. 41
3.2 The Method of Conditional Probabilities ..................... 42
3.3 Approximation Algorithms for MAXEkSAT, MAXSAT and
MAXLINEQ3-2 ............................................ 44
3.4 Approximation Algorithms for Linear Integer Programs ........ 46
3.5 Reduction of the Size of the Probability Space ................ 53
3.6 Another Approximation Algorithm for MAxE3SAT ............ 55
3.7 A PRAM Algorithm for MAXIMALINDEPENDENTSET .......... 56
X Table of Contents
.
.
Proof Checking and Non-Approximability ................... 63
Stefan Hougardy
4.1 Introduction .............................................. 63
4.2 Probabilistically Checkable Proofs ........................... 63
4.3 PCP and Non-Approximability .............................. 66
4.4 Non-Approximability of APX-Complete Problems ............. 69
4.5 Expanders and the Hardness of Approximating MAxE3SAT-b... 71
4.6 Non-Approximability of MAXCLIQUE ........................ 73
4.7 Improved Non-Approximability Results for MAXCLIQUE ........ 77
Proving the PCP-Theorem .................................. 83
Volker Heun, Woffgang Merkle, Ulrich Weigand
5.1 Introduction and Overview ................................. 83
5.2 Extended and Constant Verifiers ............................ 85
5.3 Encodings ................................................ 88
5.4 Efficient Solution Verifiers .................................. 109
5.5 Composing Verifiers and the PCP-Theorem ................... 121
5.6 The LFKN Test ........................................... 134
5.7 The Low Degree Test ...................................... 141
5.8 A Proof of Cook's Theorem ................................. 157
.
Parallel
Repetition of MIP(2,1) Systems ..................... 161
Clemens Gr6pl, Martin Skutella
6.1 Prologue ................................................. 161
6.2 Introduction .............................................. 161
6.3 Two-Prover One-Round Proof Systems ....................... 162
6.4 Reducing the Error Probability .............................. 164
6.5 Coordinate Games ......................................... 166
6.6 How to Prove the Parallel Repetition Theorem (I) ............. 168
6.7 How to Prove the Parallel Repetition Theorem (II) ............ 171
Table of Contents xl
7. Bounds for Approximating MAxLINEQ3-2 and MAxEkSAT...179
Sebastian Seibert, Thomas Wilke
7.1 Introduction .............................................. 179
7.2 Overall Structure of the Proofs .............................. 180
7.3 Long Code, Basic Tests, and Fourier Transform ............... 181
7.4 Using the Parallel Repetition Theorem for Showing Satisfiability 189
7.5 An Optimal Lower Bound for Approximating MAXLINEQ3-2 ... 192
7.6 Optimal Lower Bounds for Approximating MAxEkSAT ......... 198
8. Deriving Non-Approximabillty Results by Reductions .......
213
Claus Rick, Hein RShrig
8.1 Introduction .............................................. 213
8.2 Constraint Satisfaction Problems ............................ 215
8.3 The Quality of Gadgets .................................... 217
8.4 Weighted vs. Unweighted Problems .......................... 221
8.5 Gadget Construction ....................................... 225
8.6 Improved Results .......................................... 230
9. Optimal Non-Approximability of MAXCLIQUE ............... 235
Martin Mundhenk, Anna Slobodov~
9.1 Introduction .............................................. 235
9.2 A PCP-System for RoBE3SAT and Its Parallel Repetition ...... 236
9.3 The Long Code and Its Complete Test ....................... 239
9.4 The Non-Approximability of MAXCLIQUE .................... 243
10.
The Hardness of Approximating Set
Cover ..................
249
Alexander Wolff
10.1 Introduction .............................................. 249
10.2 The Multi-Prover Proof System ............................. 252
10.3 Construction of a Partition System .......................... 255
10.4 Reduction to Set Cover .................................... 256
11.
Semideflnite Programming and its Applications to
Approximation Algorithms .................................. 263
Thomas Hofrneister, Martin Hfihne
11.1 Introduction .............................................. 263
Xll Table of Contents
11.2 Basics from Matrix Theory ................................. 267
11.3 Semidefinite Programming .................................. 270
11.4 Duality and an Interior-Point Method ........................ 274
11.5 Approximation Algorithms for MAXCUT ..................... 278
11.6 Modeling Asymmetric Problems ............................. 285
11.7 Combining Semidefinite Programming with
Classical Approximation Algorithms ......................... 289
11.8 Improving the Approximation Ratio ......................... 293
11.9 Modeling Maximum Independent Set as a Semidefinite Program? 296
12. Dense Instances of Hard Optimization Problems ............. 299
Katja Wolf
12.1 Introduction .............................................. 299
12.2 Motivation and Preliminaries ............................... 300
12.3 Approximating Smooth Integer Programs ..................... 304
12.4 Polynomial Time Approximation Schemes for Dense MAXCUT
and MAXEkSAT Problems .................................. 308
12.5 Related Work ............................................. 310
13. Polynomial Time Approximation Schemes for Geometric
Optimization Problems in Euclidean Metric Spaces .......... 313
Richard Mayr, Annette Schelten
13.1 Introduction .............................................. 313
13.2 Definitions and Notations ................................... 314
13.3 The Structure Theorem and Its Proof ........................ 316
13.4 The Algorithm ............................................ 320
13.5 Applications to Some Related Problems ...................... 321
13.6 The Earlier PTAS ......................................... 322
Bibliography .....................................................
325
Author Index .................................................... 335
Subject Index ...................................................
337
List of Contributors .............................................
343