We study the problem of computing on large datasets that are stored on an untrusted server. We follow the approach of amortized verifiable computation introduced by Gennaro, Gentry, and Parno in CRYPTO 2010. We present the first practical verifiable computation scheme for high degree polynomial functions. Such functions can be used, for example, to make predictions based on polynomials fitted to a large number of sample points in an experiment. In addition to the many non-cryptographic applications of delegating high degree polynomials, we use our verifiable computation scheme to obtain new solutions for verifiable keyword search, and proofs of retrievability. Our constructions are based on the DDH assumption and its variants, and achieve adaptive security, which was left as an open problem by Gennaro et al (albeit for general functionalities).Our second result is a primitive which we call a verifiable database (VDB). Here, a weak client outsources a large table to an untrusted server, and makes retrieval and update queries. For each query, the server provides a response and a proof that the response was computed correctly. The goal is to minimize the resources required by the client. This is made particularly challenging if the number of update queries is unbounded. We present a VDB scheme based on the hardness of the subgroup membership problem in composite order bilinear groups. This is the first such construction that relies on a "constant-size" assumption, and does not require expensive generation of primes per operation.

Recently Raghavendra and Tan (SODA 2012) gave a 0.85-approximation algorithm for the Max Bisection problem. We improve their algorithm to a 0.8776-approximation. As Max Bisection is hard to approximate within α GW + ≈ 0.8786 under the Unique Games Conjecture (UGC), our algorithm is nearly optimal. We conjecture that Max Bisection is approximable within α GW − , i.e., that the bisection constraint (essentially) does not make Max Cut harder.We also obtain an optimal algorithm (assuming the UGC) for the analogous variant of Max 2-Sat. Our approximation ratio for this problem exactly matches the optimal approximation ratio for Max 2-Sat, i.e., α LLZ + ≈ 0.9401, showing that the bisection constraint does not make Max 2-Sat harder. This improves on a 0.93-approximation for this problem due to Raghavendra and Tan.

We consider the problem of approximating fixed-predicate constraint satisfaction problems (MAX k-CSP q (P)), where the variables take values from [q] = {0, 1,. .. , q − 1}, and each constraint is on k variables and is defined by a fixed k-ary predicate P. Familiar problems like MAX 3-SAT and MAX-CUT belong to this category. Austrin and Mossel recently identified a general class of predicates P for which MAX k-CSP q (P) is hard to approximate. They study predicates P : [q] k → {0, 1} such that the set of assignments accepted by P contains the support of a balanced pairwise independent distribution over the domain of the inputs. We refer to such predicates as promising. Austrin and Mossel show that for any promising predicate P, the problem MAX k-CSP q (P) is Unique-Games-hard to approximate better than the trivial approximation obtained by a random assignment. We give an unconditional analogue of this result in a restricted model of computation. We consider the hierarchy of semidefinite relaxations of MAX k-CSP q (P) obtained by augmenting the canonical semidefinite relaxation with the Sherali-Adams hierarchy. We show that for any promising predicate P, the integrality gap remains the same as the approximation ratio achieved by a random assignment, even after Ω(n) levels of this hierarchy.

Abstract. A predicate P : {−1, 1}k → {0, 1} can be associated with a constraint satisfaction problem Max CSP(P ). P is called "approximation resistant" if Max CSP(P ) cannot be approximated better than the approximation obtained by choosing a random assignment, and "approximable" otherwise. This classification of predicates has proved to be an important and challenging open problem. Motivated by a recent result of Austrin and Mossel (Computational Complexity, 2009), we consider a natural subclass of predicates defined by signs of quadratic polynomials, including the special case of predicates defined by signs of linear forms, and supply algorithms to approximate them as follows.In the quadratic case we prove that every symmetric predicate is approximable. We introduce a new rounding algorithm for the standard semidefinite programming relaxation of Max CSP(P ) for any predicateand analyze its approximation ratio. Our rounding scheme operates by first manipulating the optimal SDP solution so that all the vectors are nearly perpendicular and then applying a form of hyperplane rounding to obtain an integral solution. The advantage of this method is that we are able to analyze the behaviour of a set of k rounded variables together as opposed to just a pair of rounded variables in most previous methods.In the linear case we prove that a predicate called "Monarchy" is approximable. This predicate is not amenable to our algorithm for the quadratic case, nor to other LP/SDP-based approaches we are aware of.

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