13 research outputs found
On Fortification of Projection Games
A recent result of Moshkovitz \cite{Moshkovitz14} presented an ingenious
method to provide a completely elementary proof of the Parallel Repetition
Theorem for certain projection games via a construction called fortification.
However, the construction used in \cite{Moshkovitz14} to fortify arbitrary
label cover instances using an arbitrary extractor is insufficient to prove
parallel repetition. In this paper, we provide a fix by using a stronger graph
that we call fortifiers. Fortifiers are graphs that have both and
guarantees on induced distributions from large subsets. We then show
that an expander with sufficient spectral gap, or a bi-regular extractor with
stronger parameters (the latter is also the construction used in an independent
update \cite{Moshkovitz15} of \cite{Moshkovitz14} with an alternate argument),
is a good fortifier. We also show that using a fortifier (in particular
guarantees) is necessary for obtaining the robustness required for
fortification.Comment: 19 page
Parallel Repetition From Fortification
The Parallel Repetition Theorem upper-bounds the value of a repeated (tensored) two prover game in terms of the value of the base game and the number of repetitions. In this work we give a simple transformation on games â âfortificationâ â and show that for fortified games, the value of the repeated game decreases perfectly exponentially with the number of repetitions, up to an arbitrarily small additive error. Our proof is combinatorial and short. As corollaries, we obtain: (1) Starting from a PCP Theorem with soundness error bounded away from 1, we get a PCP with arbitrarily small constant soundness error. In particular, starting with the combinatorial PCP of Dinur, we get a combinatorial PCP with low error. The latter can be used for hardness of approximation as in the work of Hastad. (2) Starting from the work of the author and Raz, we get a projection PCP theorem with the smallest soundness error known today. The theorem yields nearly a quadratic improvement in the size compared to previous work. We then discuss the problem of derandomizing parallel repetition, and the limitations of the fortification idea in this setting. We point out a connection between the problem of derandomizing parallel repetition and the problem of composition. This connection could shed light on the so-called Projection Games Conjecture, which asks for projection PCP with minimal error.National Science Foundation (U.S.) (Grant 1218547
Relaxed locally correctable codes with nearly-linear block length and constant query complexity
Locally correctable codes (LCCs) are codes C: ÎŁk â ÎŁn which admit local algorithms that can correct any individual symbol of a corrupted codeword via a minuscule number of queries. One of the central problems in algorithmic coding theory is to construct O(1)-query LCC with minimal block length. Alas, state-of-the-art of such codes requires exponential block length to admit O(1)-query algorithms for local correction, despite much attention during the last two decades.
This lack of progress prompted the study of relaxed LCCs, which allow the correction algorithm to abort (but not err) on small fraction of the locations. This relaxation turned out to allow constant-query correction algorithms for codes with polynomial block length. Specifically, prior work showed that there exist O(1)-query relaxed LCCs that achieve nearly-quartic block length n = k4+α, for an arbitrarily small constant α > 0.
We construct an O(1)-query relaxed LCC with nearly-linear block length n = k1+α, for an arbitrarily small constant α > 0. This significantly narrows the gap between the lower bound which states that there are no O(1)-query relaxed LCCs with block length n = k1+o(1). In particular, this resolves an open problem raised by Gur, Ramnarayan, and Rothblum (ITCS 2018)
Composition of low-error 2-query PCPs using decodable PCPs â
The main result of this paper is a generic composition theorem for low error two-query probabilistically checkable proofs (PCPs). Prior to this work, composition of PCPs was wellunderstood only in the constant error regime. Existing composition methods in the low error regime were non-modular (i.e., very much tailored to the specific PCPs that were being composed), resulting in complicated constructions of PCPs. Furthermore, until recently, composition in the low error regime suffered from incurring an extra âconsistency â query, resulting in PCPs that are not âtwo-query â and hence, much less useful for hardness-of-approximation reductions. In a recent breakthrough, Moshkovitz and Raz [In Proc. 49th IEEE Symp. on Foundations of Comp. Science (FOCS), 2008] constructed almost linear-sized low-error 2-query PCPs for every language in NP. Indeed, the main technical component of their construction is a novel composition of certain specific PCPs. We give a modular and simpler proof of their result by repeatedly applying the new composition theorem to known PCP components. To facilitate the new modular composition, we introduce a new variant of PCP, which we call a decodable PCP (dPCP). A dPCP is an encoding of an NP witness that is both locally checkable and locally decodable. The dPCP verifier in addition to verifying the validity of the given proof like a standard PCP verifier, also locally decodes the original NP witness. Our composition is generic in the sense that it works regardless of the way the component PCPs are constructed
Rigid Matrices From Rectangular PCPs
We introduce a variant of PCPs, that we refer to as rectangular PCPs, wherein
proofs are thought of as square matrices, and the random coins used by the
verifier can be partitioned into two disjoint sets, one determining the row of
each query and the other determining the column.
We construct PCPs that are efficient, short, smooth and (almost-)rectangular.
As a key application, we show that proofs for hard languages in ,
when viewed as matrices, are rigid infinitely often. This strengthens and
simplifies a recent result of Alman and Chen [FOCS, 2019] constructing explicit
rigid matrices in FNP. Namely, we prove the following theorem:
- There is a constant such that there is an FNP-machine
that, for infinitely many , on input outputs matrices
with entries in that are -far (in Hamming distance)
from matrices of rank at most .
Our construction of rectangular PCPs starts with an analysis of how
randomness yields queries in the Reed--Muller-based outer PCP of Ben-Sasson,
Goldreich, Harsha, Sudan and Vadhan [SICOMP, 2006; CCC, 2005]. We then show how
to preserve rectangularity under PCP composition and a smoothness-inducing
transformation. This warrants refined and stronger notions of rectangularity,
which we prove for the outer PCP and its transforms.Comment: 36 pages, 3 figure
Recommended from our members
Complexity Theory
Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness, and quantum computation. Many of the developements are related to diverse mathematical ïŹelds such as algebraic geometry, combinatorial number theory, probability theory, quantum mechanics, representation theory, and the theory of error-correcting codes
Derandomized Parallel Repetition via Structured PCPs
A PCP is a proof system for NP in which the proof can be checked by a
probabilistic verifier. The verifier is only allowed to read a very small
portion of the proof, and in return is allowed to err with some bounded
probability. The probability that the verifier accepts a false proof is called
the soundness error, and is an important parameter of a PCP system that one
seeks to minimize. Constructing PCPs with sub-constant soundness error and, at
the same time, a minimal number of queries into the proof (namely two) is
especially important due to applications for inapproximability.
In this work we construct such PCP verifiers, i.e., PCPs that make only two
queries and have sub-constant soundness error. Our construction can be viewed
as a combinatorial alternative to the "manifold vs. point" construction, which
is the only construction in the literature for this parameter range. The
"manifold vs. point" PCP is based on a low degree test, while our construction
is based on a direct product test. We also extend our construction to yield a
decodable PCP (dPCP) with the same parameters. By plugging in this dPCP into
the scheme of Dinur and Harsha (FOCS 2009) one gets an alternative construction
of the result of Moshkovitz and Raz (FOCS 2008), namely: a construction of
two-query PCPs with small soundness error and small alphabet size.
Our construction of a PCP is based on extending the derandomized direct
product test of Impagliazzo, Kabanets and Wigderson (STOC 09) to a derandomized
parallel repetition theorem. More accurately, our PCP construction is obtained
in two steps. We first prove a derandomized parallel repetition theorem for
specially structured PCPs. Then, we show that any PCP can be transformed into
one that has the required structure, by embedding it on a de-Bruijn graph