8 research outputs found
Imperfect Gaps in Gap-ETH and PCPs
We study the role of perfect completeness in probabilistically checkable proof systems (PCPs) and give a way to transform a PCP with imperfect completeness to one with perfect completeness, when the initial gap is a constant. We show that PCP_{c,s}[r,q] subseteq PCP_{1,s\u27}[r+O(1),q+O(r)] for c-s=Omega(1) which in turn implies that one can convert imperfect completeness to perfect in linear-sized PCPs for NP with a O(log n) additive loss in the query complexity q. We show our result by constructing a "robust circuit" using threshold gates. These results are a gap amplification procedure for PCPs, (when completeness is not 1) analogous to questions studied in parallel repetition [Anup Rao, 2011] and pseudorandomness [David Gillman, 1998] and might be of independent interest.
We also investigate the time-complexity of approximating perfectly satisfiable instances of 3SAT versus those with imperfect completeness. We show that the Gap-ETH conjecture without perfect completeness is equivalent to Gap-ETH with perfect completeness, i.e. MAX 3SAT(1-epsilon,1-delta), delta > epsilon has 2^{o(n)} algorithms if and only if MAX 3SAT(1,1-delta) has 2^{o(n)} algorithms. We also relate the time complexities of these two problems in a more fine-grained way to show that T_2(n) <= T_1(n(log log n)^{O(1)}), where T_1(n),T_2(n) denote the randomized time-complexity of approximating MAX 3SAT with perfect and imperfect completeness respectively
On the Sensitivity Conjecture for Read-k Formulas
Various combinatorial/algebraic parameters are used to quantify the complexity of a Boolean function. Among them, sensitivity is one of the simplest and block sensitivity is one of the most useful. Nisan (1989) and Nisan and Szegedy (1991) showed that block sensitivity and several other parameters, such as certificate complexity, decision tree depth, and degree over R, are all polynomially related to one another. The sensitivity conjecture states that there is also a polynomial relationship between sensitivity and block sensitivity, thus supplying the "missing link".
Since its introduction in 1991, the sensitivity conjecture has remained a challenging open question in the study of Boolean functions. One natural approach is to prove it for special classes of functions. For instance, the conjecture is known to be true for monotone functions, symmetric functions, and
functions describing graph properties.
In this paper, we consider the conjecture for Boolean functions computable by read-k formulas. A read-k formula is a tree in which each variable appears at most k times among the leaves and has Boolean gates at its internal nodes. We show that the sensitivity conjecture holds for read-once formulas with gates computing symmetric functions. We next consider regular formulas with OR and AND gates. A formula is regular if it is a leveled tree with all gates at a given level having the same fan-in and computing the same function. We prove the sensitivity conjecture for constant depth regular read-k formulas for constant k
Communication-Rounds Tradeoffs for Common Randomness and Secret Key Generation
We study the role of interaction in the Common Randomness Generation (CRG)
and Secret Key Generation (SKG) problems. In the CRG problem, two players,
Alice and Bob, respectively get samples and
with the pairs , , being drawn independently
from some known probability distribution . They wish to communicate so as
to agree on bits of randomness. The SKG problem is the restriction of the
CRG problem to the case where the key is required to be close to random even to
an eavesdropper who can listen to their communication (but does not have access
to the inputs of Alice and Bob). In this work, we study the relationship
between the amount of communication and the number of rounds of interaction in
both the CRG and the SKG problems. Specifically, we construct a family of
distributions , parametrized by integers , and ,
such that for every there exists a constant for which CRG
(respectively SKG) is feasible when with
rounds of communication, each consisting of bits, but when
restricted to rounds of interaction, the total communication must
exceed bits. Prior to our work no separations were
known for .Comment: 41 pages, 3 figure
Optimal Fine-Grained Hardness of Approximation of Linear Equations
The problem of solving linear systems is one of the most fundamental problems
in computer science, where given a satisfiable linear system , for and , we wish to find a vector
such that . The current best algorithms for
solving dense linear systems reduce the problem to matrix multiplication, and
run in time . We consider the problem of finding
-approximate solutions to linear systems with respect to the
-norm, that is, given a satisfiable linear system , find an such that . Our main result is a fine-grained reduction
from computing the rank of a matrix to finding -approximate
solutions to linear systems. In particular, if the best known
time algorithm for computing the rank of matrices is optimal
(which we conjecture is true), then finding an -approximate
solution to a dense linear system also requires
time, even for as large as . We also
prove (under some modified conjectures for the rank-finding problem) optimal
hardness of approximation for sparse linear systems, linear systems over
positive semidefinite matrices, well-conditioned linear systems, and
approximately solving linear systems with respect to the -norm, for . At the heart of our results is a novel reduction from the rank problem
to a decision version of the approximate linear systems problem. This reduction
preserves properties such as matrix sparsity and bit complexity.Comment: To appear in ICALP 202