In this paper, we present improved inapproximability re-sults for three problems: the problem of finding the maximum clique size in a graph, the problem of finding the chro-matic number of a graph, and the problem of coloring a graph with a small chromatic number with a small numberof colors. H*astad's celebrated result [13] shows that the maximumclique size in a graph with n vertices is inapproximable inpolynomial time within a factor n1-ffl for arbitrarily smallconstant ffl> 0 unless NP=ZPP. In this paper, we aimat getting the best subconstant value of ffl in H*astad's re-sult. We prove that clique size is inapproximable within a factor n2(log n)1-fl (corresponding to ffl = 1(log n)fl) for some constant fl> 0 unless NP ` ZPTIME(2(log n) O(1)). This improves the previous best inapproximability factor of

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For every integer k > 0, we present a PCP characterization of NP where the verifier uses logarithmic randomness, queries 4k + k 2 bits in the proof, accepts a correct proof with probability 1 (i.e. it is has perfect completeness) and accepts any supposed proof of a false statement with probability at most 2 k 2 +1 . In particular, the verifier achieves optimal amortized query complexity of 1 + for arbitrarily small constant > 0. Such a characterization was already proved by Samorodnitsky and Trevisan [15], but their verifier loses perfect completeness and their proof makes an essential use of this feature. By using an adaptive verifier we can decrease the number of query bits to 2k + k 2 , the same number obtained in [15]. Finally we extend some of the results to larger domains. Royal Institute of Technology, Stockholm, work done while visiting Institute for Advanced Study, supported by NSF grant CCR-9987077.

We show that the N P-Complete language 3SAT has a PCP verifier that makes two queries to a proof of almost-linear size and achieves sub-constant probability of error o(1). The verifier performs only projection tests, meaning that the answer to the first query determines at most one accepting answer to the second query. Previously, by the parallel repetition theorem, there were PCP Theorems with two-query projection tests, but only (arbitrarily small) constant error and polynomial size [29]. There were also PCP Theorems with sub-constant error and almost-linear size, but a constant number of queries that is larger than 2 [26]. As a corollary, we obtain a host of new results. In particular, our theorem improves many of the hardness of approximation results that are proved using the parallel repetition theorem. A partial list includes the following: 1. 3SAT cannot be efficiently approximated to within a factor of 7 8 + o(1), unless P = N P. This holds even under almost-linear reductions. Previously, the best known N P-hardness

For Boolean functions that are ɛ-far from the set of linear functions, we study the lower bound on the rejection probability (denoted by rej(ɛ)) of the linearity test suggested by Blum, Luby and Rubinfeld. This problem is arguably the most fundamental and extensively studied problem in property testing of Boolean functions. The previously best bounds for rej(ɛ) were obtained by Bellare, Coppersmith, H˚astad, Kiwi and Sudan. They used Fourier analysis to show that rej(ɛ) ≥ ɛ for every 0 ≤ ɛ ≤ 1/2. They also conjectured that this bound might not be tight for ɛ’s which are close to 1/2. In this paper we show that this indeed is the case. Specifically, we improve the lower bound of rej(ɛ) ≥ ɛ by an additive constant that depends only on ɛ: rej(ɛ) ≥ ɛ + min{1376ɛ 3 (1 − 2ɛ) 12, 1 4 ɛ(1 − 2ɛ)4}, for every 0 ≤ ɛ ≤ 1/2. Our analysis is based on a relationship between rej(ɛ) and the weight distribution of a coset code of the Hadamard code. We use both Fourier analysis and coding theory tools to estimate this weight distribution.

Abstract. In this paper, we study two questions related to the problem of testingwhether a function is close to a homomorphism. For two finite groups

Motivated by the pervasiveness of strong inapproximability results for Max-CSPs, we introduce a relaxed notion of an approximate solution of a Max-CSP. In this relaxed version, loosely speaking, the algorithm is allowed to replace the constraints of an instance by some other (possibly real-valued) constraints, and then only needs to satisfy as many of the new constraints as possible. To be more precise, we introduce the following notion of a predicate P being useful for a (realvalued) objective Q: given an almost satisfiable Max-P instance, there is an algorithm that beats a random assignment on the corresponding Max-Q instance applied to the same sets of literals. The standard notion of a nontrivial approximation algorithm for a Max-CSP with predicate P is exactly the same as saying that P is useful for P itself. We say that P is useless if it is not useful for any Q. This turns out to be equivalent to the following pseudo-randomness property: given an almost satisfiable instance of Max-P it is hard to find an assignment such that the induced distribution on k-bit strings defined by the instance is not essentially uniform. Under the Unique Games Conjecture, we give a complete and simple characterization of useful Max-CSPs defined by a predicate: such a Max-CSP is useless if and only if there is a pairwise independent distribution supported on the satisfying assignments of the predicate. It is natural to also consider the case when no negations are allowed in the CSP instance, and we derive a similar complete characterization (under the UGC) there as well. Finally, we also include some results and examples shedding additional light on the approximability of certain Max-CSPs. Funded by NSERC.