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.

/pdf/a-threshold-of-ln-n-for-approximating-set-cover-3tgfinxlup.pdf

A threshold of ln n for approximating set cover

Our main result is a reduction from worst-case lattice problems such as SVP and SIVP to a certain learning problem. This learning problem is a natural extension of the 'learning from parity with error' problem to higher moduli. It can also be viewed as the problem of decoding from a random linear code. This, we believe, gives a strong indication that these problems are hard. Our reduction, however, is quantum. Hence, an efficient solution to the learning problem implies a quantum algorithm for SVP and SIVP. A main open question is whether this reduction can be made classical.Using the main result, we obtain a public-key cryptosystem whose hardness is based on the worst-case quantum hardness of SVP and SIVP. Previous lattice-based public-key cryptosystems such as the one by Ajtai and Dwork were only based on unique-SVP, a special case of SVP. The new cryptosystem is much more efficient than previous cryptosystems: the public key is of size O(n2) and encrypting a message increases its size by O(n)(in previous cryptosystems these values are O(n4) and O(n2), respectively). In fact, under the assumption that all parties share a random bit string of length O(n2), the size of the public key can be reduced to O(n).

/pdf/on-lattices-learning-with-errors-random-linear-codes-and-2dvfl9suqh.pdf

On lattices, learning with errors, random linear codes, and cryptography

We study the role that privacy-preserving algorithms, which prevent the leakage of specific information about participants, can play in the design of mechanisms for strategic agents, which must encourage players to honestly report information. Specifically, we show that the recent notion of differential privacv, in addition to its own intrinsic virtue, can ensure that participants have limited effect on the outcome of the mechanism, and as a consequence have limited incentive to lie. More precisely, mechanisms with differential privacy are approximate dominant strategy under arbitrary player utility functions, are automatically resilient to coalitions, and easily allow repeatability. We study several special cases of the unlimited supply auction problem, providing new results for digital goods auctions, attribute auctions, and auctions with arbitrary structural constraints on the prices. As an important prelude to developing a privacy-preserving auction mechanism, we introduce and study a generalization of previous privacy work that accommodates the high sensitivity of the auction setting, where a single participant may dramatically alter the optimal fixed price, and a slight change in the offered price may take the revenue from optimal to zero.

/pdf/mechanism-design-via-differential-privacy-z5vjyvpisy.pdf

Mechanism Design via Differential Privacy

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

/pdf/expander-graphs-and-their-applications-wbsb8gxfw2.pdf

Expander graphs and their applications

We give a new characterization of NP: the class NP contains exactly those languages L for which membership proofs (a proof that an input x is in L) can be verified probabilistically in polynomial time using logarithmic number of random bits and by reading sublogarithmic number of bits from the proof.We discuss implications of this characterization; specifically, we show that approximating Clique and Independent Set, even in a very weak sense, is NP-hard.

Probabilistic checking of proofs: a new characterization of NP

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

The contribution of this paper is two-fold. First, a connection is established between approximating the size of the largest clique in a graph and multi-prover interactive proofs. Second, an efficient multi-prover interactive proof for NP languages is constructed, where the verifier uses very few random bits and communication bits. Last, the connection between cliques and efficient multi-prover interaction proofs, is shown to yield hardness results on the complexity of approximating the size of the largest clique in a graph.Of independent interest is our proof of correctness for the multilinearity test of functions.

https://dl.acm.org/doi/pdf/10.1145/226643.226652

Interactive proofs and the hardness of approximating cliques

The authors give a new characterization of NP: the class NP contains exactly those languages L for which membership proofs (a proof that an input x is in L) can be verified probabilistically in polynomial time using logarithmic number of random bits and sub-logarithmic number of queries to the proof. This is a non-relativizing characterization of NP. They discuss implications of this characterization; specifically, they show that approximating clique (or independent set) is NP-hard. >

Probabilistic checking of proofs; a new characterization of NP

The paper considers the computational hardness of approximating the chromatic number of a graph. The authors first give a simple proof that approximating the chromatic number of a graph to within a constant power (of the value itself) in NP-hard. They then consider the hardness of coloring a 3-colorable graph with as few as possible colors. They show that determining whether a graph is 3-colorable or any legal coloring of it requires at least 5 colors is NP-hard. Therefore, coloring a 3-colorable graph with 4 colors is NP-hard. >

Shmuel Safra

Papers

Probabilistic checking of proofs: a new characterization of NP

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

Interactive proofs and the hardness of approximating cliques

Probabilistic checking of proofs; a new characterization of NP

On the hardness of approximating the chromatic number