This beginning graduate textbook describes both recent achievements and classical results of computational complexity theory. Requiring essentially no background apart from mathematical maturity, the book can be used as a reference for self-study for anyone interested in complexity, including physicists, mathematicians, and other scientists, as well as a textbook for a variety of courses and seminars. More than 300 exercises are included with a selected hint set.

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Computational Complexity: A Modern Approach

Given a collection ℱ of subsets of S = {1,…,n}, set cover is the problem of selecting as few as possible subsets from ℱ such that their union covers S,, and max k-cover is the problem of selecting k subsets from ℱ such that their union has maximum cardinality. Both these problems are NP-hard. We prove that (1 - o(1)) ln n is a threshold below which set cover cannot be approximated efficiently, unless NP has slightly superpolynomial time algorithms. This closes the gap (up to low-order terms) between the ratio of approximation achievable by the greedy alogorithm (which is (1 - o(1)) ln n), and provious results of Lund and Yanakakis, that showed hardness of approximation within a ratio of (log2n) / 2 ≃0.72 ln n. For max k-cover, we show an approximation threshold of (1 - 1/e)(up to low-order terms), under assumption that P ≠ NP.

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A threshold of ln n for approximating set cover

A major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists, since expander graphs, the protagonists of our story, come up in numerous and often surprising contexts in both fields But, perhaps, we should start with a few words about graphs in general They are, of course, one of the prime objects of study in Discrete Mathematics However, graphs are among the most ubiquitous models of both natural and human-made structures In the natural and social sciences they model relations among species, societies, companies, etc In computer science, they represent networks of communication, data organization, computational devices as well as the flow of computation, and more In mathematics, Cayley graphs are useful in Group Theory Graphs carry a natural metric and are therefore useful in Geometry, and though they are “just” one-dimensional complexes, they are useful in certain parts of Topology, eg Knot Theory In statistical physics, graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such systems The study of these models calls, then, for the comprehension of the significant structural properties of the relevant graphs But are there nontrivial structural properties which are universally important? Expansion of a graph requires that it is simultaneously sparse and highly connected Expander graphs were first defined by Bassalygo and Pinsker, and their existence first proved by Pinsker in the early ’70s The property of being an expander seems significant in many of these mathematical, computational and physical contexts It is not surprising that expanders are useful in the design and analysis of communication networks What is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness In mathematics, we will encounter eg their role in the study of metric embeddings, and in particular in work around the Baum-Connes Conjecture Expansion is closely related to the convergence rates of Markov Chains, and so they play a key role in the study of Monte-Carlo algorithms in statistical mechanics and in a host of practical computational applications The list of such interesting and fruitful connections goes on and on with so many applications we will not even

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Expander graphs and their applications

In this chapter we survey the theory of two-party communication complexity. This field of theoretical computer science aims at studying the following, seemingly very simple, scenario: There are two players Alice who holds an n-bit string x and Bob who holds an n-bit string y. Their goal is to communicate in order to compute the value of some boolean function f(x, y), while exchanging a number of bits which is as small as possible. In the first part of this survey we present, mainly by giving examples, some of the results (and techniques) developed as part of this theory. We put an emphasis on proving lower bounds on the amount of communication that must be exchanged in the above scenario for certain functions f . In the second part of this survey we will exemplify the wide applicability of the results proved in the first part to other areas of computer science. While it is obvious that there are many applications of the results to problems in which communication is involved (e.g., in distributed systems), we concentrate on applications in which communication does not appear explicitly in the statement of the problems. In particular, we present results regarding the following models of computation: • Finite automata • Turing machines • Decision trees • Ordered binary decision diagrams (OBDDs) • VLSI chips • Networks of threshold gates We provide references to many other issues and applications of communication complexity which are not discussed in this survey.

Communication Complexity

Usually, a proof of a theorem contains more knowledge than the mere fact that the theorem is true. For instance, to prove that a graph is Hamiltonian it suffices to exhibit a Hamiltonian tour in it; however, this seems to contain more knowledge than the single bit Hamiltonian/non-Hamiltonian.In this paper a computational complexity theory of the “knowledge” contained in a proof is developed. Zero-knowledge proofs are defined as those proofs that convey no additional knowledge other than the correctness of the proposition in question. Examples of zero-knowledge proof systems are given for the languages of quadratic residuosity and 'quadratic nonresiduosity. These are the first examples of zero-knowledge proofs for languages not known to be efficiently recognizable.

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The knowledge complexity of interactive proof-systems

We introduce a new low-degree-test, one that uses the restriction of low-degree polynomials to planes (i. e., afine sub-spaces of dimension 2), rather than the restriction to lines (i. e., afine sub-spaces of dimension 1). We prove the new test to be of a very small emorprobability (in particular, much smaller than constant). The new test enables us to prove a low-error characterization of NP in terms of PCP. Specifically, OUT theorem states that, for any given c > 0, membership in any NP language can be verijied with 0(1) accesses, each r’eading logarithmic number of bits, and such that the error-probability is 2‘“~’-’ n. Our results are in fact stronger, as stated below. One application of the new characterization of NP is that approximating SETCOVER to within a logarithmic factors is NP-hard. Previous analysis for low-degree-tests, as well as previous characten”zations of NP in terms of PCP, have managed to achieve, with constant number of accesses, error-probability of, at best, a constant. The proof for the smail err-or-probability of our new low-degree-test is, nevertheless, significantly simpler than previous proofs. In particular, it is combinatorial and geometrical in nature, rather than algebraic. 1 Characterizations of NP in terms of PCP Characterizing the class NP, by itself or with respect to other computational-complexity clssses, is perhaps one *URL http: //www.math.tau. ac.il/school/courses/PCP tWejzmm ~st., ISRAEL. ~-a@wj~dom.wej--$ ac,J 1Tel-Aviv University, ISRAEL. Permissionto nmhcdigiiol/hmi copies ol’iill w poti ,d’hs motcrizl Ibr personal or claswoom use is grmltd without fee provided Ih:)t the copies ‘are nof m.adt or disirihutcd I“or pI-o~II or ccrmmcrc. ial :Id\’Jm:Igt. Ilw L’OPV right notice. the Iil!c ot’lhc puhiimlion nnd iLs tiate nppcm’, imd uotice is given that copyright is by pcnniwi(m oflhe AChl, 10C. To copyolherwise. 10rcpuhlish. 10 POM m swvws or In redis(rilul!c In lists. requires specilic permission amikrr I’te .STOC ‘ 97 El ]>;lSO. ‘] ’~XSS [ IS.A Copyrighl1997 ACM O-8979 I-XNL61’9705 ,$3.50 of the most fundamental avenues of research in theory of computer-science. Since the early days, when the classes P and NP were defined, and the question was posed as to whether they are the same or do they differ, many problems were shown to be NP-complete, thereby increasing the weight on finding stricter characterization for the class NP. NP has since been given a few alternative characterizations. The one most commonly applied being Cook’s [CO071], which characterizes NP in terms of efficient verification of proofs (or nondeterministic computations). A new perspective, by which improved characterizations of NP can be obtained, has been recently proposed. The motivation for which stems from questions regarding the hardness of approximation versions, for problems whose exact computation is known to be NP-hard. This avenue of research was initiated by [FGL+91], which introduced a new methodology for proving hardness results for approximation problems. The method takes advantage of results in a seemingly unrelated area — that of interactive proofs [GMR89, Bab85, BGKW88, LFKN92, Sha92, BFL91] — however interprets those results with quite a different perspective in mind. Much effort has been invested since towards a better understanding of this methodology, and the class NP has consequently gained stricter characterizations [AS92, ALM+92, BGS95], which are referred to as characterizations of NP in terms of PCP (or, in short, PCP characterizations of NP). The PCP characterization of NP — though has taken around 20 years to be formulated — seems now as the most natural extension of the old characterization of NP [CO071], if one has in mind proving hardness results for approximation problems. This characterization has already been used to obtain quite a few hardness results for approximation problems [FGL+91, AS92, ALM+ 92, PY91, LY94, BGLR93, KLS93, BGS95, Hiis96a, H5a96b, H5s97]. The previous characterization of NP in terms of the PCP hierarchy [AS92, ALM+ 92], seemed at first ss the best possible up to constant factors. A stronger characterization was later conjectured in [BGLR93]; one that, as an outcome, implies NP-hardness of approximating SET-COVER to within logarithmic factors [LY94, BGLR93]. The conjecture itself, more-

A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP

We show that a parallel repetition of any two-prover one-round proof system (MIP(2,1)) decreases the probability of error at an exponential rate. No constructive bound was previously known. The constant in the exponent (in our analysis) depends only on the original probability of error and on the total number of possible answers of the two provers. The dependency on the total number of possible answers is logarithmic, which was recently proved to be almost the best possible [U. Feige and O. Verbitsky, Proc.11th Annual IEEE Conference on Computational Complexity, IEEE Computer Society Press, Los Alamitos, CA, 1996, pp. 70--76].

A Parallel Repetition Theorem

We show that a parallel repetition of any two-prover one-round proof system (MIP(2; 1)) decreases the probability of error at an exponential rate. No constructive bound was previously known. The constant in the exponent (in our analysis) depends only on the original probability of error and on the total number of possible answers of the two provers. The dependency on the total number of possible answers is logarithmic, which was recently proved to be almost the best possible (U. Feige and O. Verbitsky, Proc. 11th Annual IEEE Conference on Computational Complexity, IEEE Computer Society Press, Los Alamitos, CA, 1996, pp. 70{76).

Ran Raz

Papers

A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP

A Parallel Repetition Theorem

A parallel repetition theorem

ProMate: a structure based prediction program to identify the location of protein-protein binding sites.

Distance labeling in graphs