Probabilistic Checking of Pro ofs and Hardness of Approximation Problems Sanjeev Arora CSTR Revised version of a dissertation submitted at CS Division UC Berkeley in August Princeton techrep orts are available in p ostscript form via anonymous ftp from ftpcsprincetonedu InterNet address You may log in as anonymous and giveyour email ad dress as password Change to the directory reports The le README gives details and the le INDEX lists the numb ers and the titles of all rep orts This rep ort is numb ered Hardcopies of this rep ort are available for a charge of only US dollar checks accepted from the following address Technical Rep orts Department of Computer Science Princeton University Olden Street Princeton NJ Probabilistic Checking of Pro ofs and Hardness of Approximation Problems c Copyright by Sanjeev Arora The dissertation committee consisted of Professor Umesh V Vazirani Chair Professor Richard M Karp Professor John W Addison The authors current address Department of Computer Science Olden St Princeton University Princeton NJ email aroracsprincetonedu Tomyparents and members of my family including the ones who just joined Contents Intro duction This dissertation Hardness of Approximation A New Characterization of NP Knowledge assumed of the Reader Old vs New Views of NP The Old View Co okLevin Theorem Optimization and Approximation A New View of NP The PCP Theorem Connection to Approximation History and Background PCPAnOverview Co des Pro of of the PCP Theorem an Outline History and Background A Pro of of the PCP Theorem Polynomial Co des and Their Use Algebraic Pro cedures for Polynomial Co des An Application Aggregating Queries AVerier Using O log n Random Bits A Less EcientVerier Checking Split Assignments AVerier using O query bits Checking Split Assignments The Algebraic Pro cedures SumCheck Pro cedures for the Linear Function Co de Pro cedures for General Polynomial Co de The Overall Picture HistoryAttributions iii The Lowdegree Test The Bivariate Case Correctness of the Lowdegree Test Discussion History Hardness of Approximations The Canonical Problems GapPreserving Reductions MAXSNP Problems on Lattices Co des Linear Systems The Problems The Results Signicance of the Results A Set of Vectors Reductions to NV and others Hardness of Approximating SV Proving n approximations NPhard MAXSATISFY Other Inapproximability Results A Survey Historical NotesFurther Reading eriers that make queries PCP V Hardness of Approximating Lab el Cover Hardness of SV Unifying Lab elCover and MAXSAT Applications of PCP Techniques Strong Forms of the PCP Theorem The Applications Exact Characterization of Nondeterministic Time Classes Transparent Math Pro ofs Checking Computations Micalis Certicates for VLSI Chips Characterization of PSPACE Condon et al Probabilistically Checkable Co des Kilians ZK arguments Khanna et als Structure Theorem for MAXSNP The Hardness of nding Small Cliques Op en Problems Hardness of Approximations Proving hardness where no results exist Improving existing hardness results Obtaining Logical Insightin to Approximation Problems Op en Problems connected with PCP techniques Do es the PCP theorem have a simpler pro of A Library of Useful Facts List of Figures Tableau and window The windowshows the nite control initially in state q and reading a The control overwrites the with a moves one cell to right and changes state to q Verier in the denition of PCP a Pro of that V exp ects b Pro of that V exp ects Shaded area in new represents the assignment split into Q parts that corresp onds to V s random seed r m A function from h to F can b e extended to a p olynomial of degree mh A table of partial sums may b e conceptualized as a tree of branching factor q The Sumcheckfollows a random path down this tree How to enco de an assignment so that the PCPlog n verier accepts with probability Lab el Cover instance for formula x x x x x x The symb ols on edge e representmap e Figure showing the epro jections of vectors V and V in the vector v a v a set where e v v vii viii Acknowledgements Imust acknowledge my great debt to my advisor Umesh Vazirani for the nurturing and advising the sake and the dinners he has provided me over the last four years My only regret is that I never to ok squash lessons from him even though I kept planning to The theory group at Berkeley wasawonderful enviroment in which to b e a graduate student Thanks to Dick Karp Manuel Blum Mike LubyRaimund Seidel Gene Lawler and Abhiram Ranade for creating the environment Dicks phenomenally clear mind proved a go o d resource on many o ccasions Manuel and Mike got me thinking ab out manyofthe problems up on which this dissertation is based Some of Mikes unpublished observations were central to guiding my dissertation research in the right direction Iwould like to thank my colleagues with whom I did this research Sp ecial thanks go to Madhu Sudan and Muli Safra for their patience while collab orating with me in my early gradscho ol days I learnt a lot from them I also thank Carsten Lund Jacques Stern Laci Babai Mario Szegedy Ra jeev Motwani and Z Sweedyk for collab orating with me Thanks to John Addison for serving on my dissertation committee and giving me lots of comments in his usual precise style Thanks to all my fellowstudents in the department and the p ostdo cs at ICSI for talks TGIFs happy hours and bullsessions Thanks also to all the p eople at MIT who got me interested in theoretical computer science and softball during my undergraduate years I am esp ecially grateful to Bruce Maggs Tom Leighton Mike Sipser Mauricio Karchmer Richard Stanley Charles Leiserson and David Shmoys There are many others who over the years have help ed me develop as a p erson and a researcher I will not attempt to list their names for fear that anysuch list will b e incomplete but I express my thanks to them all My deep est gratitude is to my family for instilling a love of knowledge in me I hop e my father is not to o dismayed that I am contributing two more professors to the family FinallyIwould like to thank Silvia She was clearly my b est discovery at Berkeley Chapter Intro duction The study of the diculty or hardness of computational problems has two parts The theory of algorithms is concerned with the design of ecient algorithmsin other w ords with proving upp er b ounds on the amount of computational resources required to solve a sp ecic problem Complexity theory is concerned with proving the corresp onding lower b ounds Our work is part of the second endeavormore sp ecically the endeavor to prove problems computationally dicult or hard Despite some notable successes lower b ound research is still in a stage of infancy progress on its op en problems has b een slow Central among these op en problems is the question whether P NP In other words is there a problem that can b e solved in p olynomial time on a nondeterministic Turing Machine but cannot b e solved in p olynomial time deterministically The conjecture P NP is widely b elieved but currently our chances of proving it app ear slim If we assume that P NP however then another interesting question arises given any sp ecic optimization problem of interest is it in P or not Complexity theory has had remarkable success in answering such questions The theory of NPcompleteness due to Co ok Levin and Karp allows us to prove that explicit problems are not in P assuming P NP The main idea is to prove the given problem NPhard that is to give a p olynomial time reduction from instances of any NP problem to instances of the given problem If an NPhard problem were to have a p olynomialtime algorithm so w ould every NP problem whichwould contradict the assumption P NP Hence if P NP then an NPhard problem has no p olynomialtime algorithm To put it dierently an NPhard problem is no easier than any other problem in NP The success of the theory of NP completeness lay in the unityitbrought to the study of computational complexity a wide array of optimization problems arising in practice and which had hitherto deed all eorts of algorithm designers to nd ecient algorithms were proved NPhard in one swo op using essentially the same kind of reductions For a list of NPhard problems c see the survey by Garey and Johnson GJ CHAPTER INTRODUCTION But one ma jor group of problems seemed not to t in the framework of NPcompleteness approximation problems Approximating an NPhard optimization problem within a factor c means to compute solutions whose cost is within a multiplicative factor c of the cost of the optimal solution Such solutions would suce in practice if c were close enough to For some NPhard problems weknowhow to compute such solutions in p olynomial timefor most it seemed that ev en approximation was hard at least a substantial b o dy of