This work addresses k-restriction problems, which unify combinatorial problems of the following type: The goal is to construct a short list of strings in Σ m that satisfies a given set of k-wise demands. For every k positions and every demand, there must be at least one string in the list that satisfies the demand at these positions. Problems of this form frequently arise in different fields in Computer Science.The standard approach for deterministically solving such problems is via almost k-wise independence or k-wise approximations for other distributions. We offer a generic algorithmic method that yields considerably smaller constructions. To this end, we generalize a previous work of Naor, Schulman and Srinivasan [18]. Among other results, we greatly enhance the combinatorial objects in the heart of their method, called splitters, and construct multi-way splitters, using a new discrete version of the topological Necklace Splitting Theorem [1].We utilize our methods to show improved constructions for group testing [19] and generalized hashing [3], and an improved inapproximability result for Set-Cover under the assumption P = N P.

We introduce and study a new model of interactive proofs: AM (k), or Arthur-Merlin with k non-communicating Merlins. Unlike with the better-known MIP, here the assumption is that each Merlin receives an independent random challenge from Arthur. One motivation for this model (which we explore in detail) comes from the close analogies between it and the quantum complexity class QMA (k), but the AM (k) model is also natural in its own right.We illustrate the power of multiple Merlins by giving an AM (2) protocol for 3Sat, in which the Merlins' challenges and responses consist of only n 1/2+o(1) bits each. Our protocol has the consequence that, assuming the Exponential Time Hypothesis (ETH), any algorithm for approximating a dense CSP with a polynomial-size alphabet must take n (log n) 1−o(1) time. Algorithms nearly matching this lower bound are known, but their running times had never been previously explained. Brandão and Harrow have also recently used our 3Sat protocol to show quasipolynomial hardness for approximating the values of certain entangled games.In the other direction, we give a simple quasipolynomial-time approximation algorithm for free games, and use it to prove that, assuming the ETH, our 3Sat protocol is essentially optimal. More generally, we show that multiple Merlins never provide more than a polynomial advantage over one: that is, AM (k) = AM for all k = poly (n). The key to this result is a subsampling theorem for free games, which follows from powerful results by Alon et al. and Barak et al. on subsampling dense CSPs, and which says that the value of any free game can be closely approximated by the value of a logarithmic-sized random subgame.

We suggest the research agenda of establishing new hardness of approximation results based on the "projection games conjecture", i.e., an instantiation of the Sliding Scale Conjecture of Bellare, Goldwasser, Lund and Russell to projection games.We pursue this line of research by establishing a tight N P-hardness result for the SetCover problem. Specifically, we show that under the projection games conjecture (in fact, under a quantitative version of the conjecture that is only slightly beyond the reach of current techniques), it is N P-hard to approximate Set-Cover on instances of size N to within (1 − α) ln N for arbitrarily small α > 0. Our reduction establishes a tight trade-off between the approximation accuracy α and the time required for the approximation 2

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 factor was 7 8 + ε for any constant ε > 0, under polynomial reductions (Håstad,[18]). 2. 3LIN cannot be efficiently approximated to within a factor of 1 2 + o(1), unless P = N P. This holds even under almost-linear reductions. Previously, the best known N P-hardness factor was 1 2 + ε for any constant ε > 0, under polynomial reductions (Håstad,[18]). 3. A PCP Theorem with amortized query complexity 1+o(1) and amortized free bit complexity o(1). Previously, the best known amortized query complexity and free bit complexity were 1 + ε and ε, respectively, for any constant ε > 0 (Samorodnitsky and Trevisan, [32]).One of the new ideas that we use is a new technique for doing the composition step in the (classical) proof of the PCP Theorem, without increasing the number of queries to the proof. We formalize this as a composition of new objects that we call Locally Decode/Reject Codes (LDRC). The notion of LDRC was implicit in several previous works, and we make it explicit in this work. We believe that the formulation of LDRCs and their construction are of independent interest.

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 factor was 7 8 + ε for any constant ε > 0, under polynomial reductions (Håstad,[18]). 2. 3LIN cannot be efficiently approximated to within a factor of 1 2 + o(1), unless P = N P. This holds even under almost-linear reductions. Previously, the best known N P-hardness factor was 1 2 + ε for any constant ε > 0, under polynomial reductions (Håstad,[18]). 3. A PCP Theorem with amortized query complexity 1+o(1) and amortized free bit complexity o(1). Previously, the best known amortized query complexity and free bit complexity were 1 + ε and ε, respectively, for any constant ε > 0 (Samorodnitsky and Trevisan, [32]).One of the new ideas that we use is a new technique for doing the composition step in the (classical) proof of the PCP Theorem, without increasing the number of queries to the proof. We formalize this as a composition of new objects that we call Locally Decode/Reject Codes (LDRC). The notion of LDRC was implicit in several previous works, and we make it explicit in this work. We believe that the formulation of LDRCs and their construction are of independent interest.