Siavosh Benabbas

Verifiable delegation of computation over large datasets

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 noncryptographic 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.

SDP Gaps from Pairwise Independence

We consider the problem of approximating fixed-predicate constraint satisfaction problems (MAX k-CSPq(P)), where the variables take values from (q) =f0; 1;:::; q 1g, 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-CSPq(P) is hard to approximate. They study predicates P : (q) k !f0; 1g 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-CSPq(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-CSPq(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 W(n) levels of this hierarchy.

Better Balance by Being Biased: A 0.8776-Approximation for Max Bisection

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 αGW + ε ≈ 0.8786 under the Unique Games Conjecture (UGC), our algorithm is nearly optimal. We conjecture that Max Bisection is approximable within αGW − ε, that is, 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, that is, αLLZ + ε ≈ 0.9401, showing that the bisection constraint does not make Max 2-Sat harder. This improves on a 0.93-approximation for this problem from Raghavendra and Tan.

Extending SDP integrality gaps to sherali-adams with applications to quadratic programming and maxcutgain

We show how under certain conditions one can extend constructions of integrality gaps for semidefinite relaxations into ones that hold for stronger systems: those SDP to which the so-called k-level constraints of the Sherali-Adams hierarchy are added. The value of k above depends on properties of the problem. We present two applications, to the Quadratic Programming problem and to the MaxCutGain problem. Our technique is inspired by a paper of Raghavendra and Steurer [Raghavendra and Steurer, FOCS 09] and our result gives a doubly exponential improvement for Quadratic Programming on another result by the same authors [Raghavendra and Steurer, FOCS 09]. They provide tight integrality-gap for the system above which is valid up to k=(loglogn)Ω(1) whereas we give such a gap for up to k=nΩ(1).

Tight Gaps for Vertex Cover in the Sherali-Adams SDP Hierarchy

We give the first tight integrality gap for Vertex Cover in the Sherali-Adams SDP system. More precisely, we show that for every \epsilon >0, the standard SDP for Vertex Cover that is strengthened with the level-6 Sherali-Adams system has integrality gap 2-\epsilon. To the best of our knowledge this is the first nontrivial tight integrality gap for the Sherali-Adams SDP hierarchy for a combinatorial problem with hard constraints. For our proof we introduce a new tool to establish Local-Global Discrepancy which uses simple facts from high-dimensional geometry. This allows us to give Sherali-Adams solutions with objective value n(1/2+o(1)) for graphs with small (2+o(1)) vector chromatic number. Since such graphs with no linear size independent sets exist, this immediately gives a tight integrality gap for the Sherali-Adams system for superconstant number of tightenings. In order to obtain a Sherali-Adams solution that also satisfies semidefinite conditions, we reduce semidefiniteness to a condition on the Taylor expansion of a reasonably simple function that we are able to establish up to constant-level SDP tightenings. We conjecture that this condition holds even for superconstant levels which would imply that in fact our solution is valid for superconstant level Sherali-Adams SDPs.

On quadratic threshold CSPs

A predicate P:{– 1, 1}k→{0,1} can be associated with a constraint satisfaction problem

Better balance by being biased: a 0.8776-approximation for Max Bisection

\textsc{Max CSP}{(P)}

The Sherali-Adams System Applied to Vertex Cover: Why Borsuk Graphs Fool Strong LPs and some Tight Integrality Gaps for SDPs.

. P is called “approximation resistant” if

Efficient Sum-Based Hierarchical Smoothing Under \ell_1-Norm

\textsc{Max CSP}{(P)}

Extending SDP Integrality Gaps to Sherali-Adams with Applications to

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