Mihir Bellare, Oded Goldreich, Madhu Sudan

This paper continues the investigation of the connection between proof

systems and approximation. The emphasis is on proving ``tight''

non-approximability results via consideration of measures like the

``free bit complexity'' and the ``amortized free bit complexity'' of

proof systems.

The first part of the paper presents a collection of new ... more >>>

Irit Dinur, Guy Kindler, Shmuel Safra

This paper shows finding the closest vector in a lattice

to be NP-hard to approximate to within any factor up to

$2^{(\log{n})^{1-\epsilon}}$ where

$\epsilon = (\log\log{n})^{-\alpha}$

and $\alpha$ is any positive constant $<{1\over 2}$.

Irit Dinur, Eldar Fischer, Guy Kindler, Ran Raz, Shmuel Safra

This paper strengthens the low-error PCP characterization of NP, coming

closer to the ultimate BGLR conjecture. Namely, we prove that witnesses for

membership in any NP language can be verified with a constant

number of accesses, and with an error probability exponentially

small in the ...
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Venkatesan Guruswami, Sanjeev Khanna

We give a new proof showing that it is NP-hard to color a 3-colorable

graph using just four colors. This result is already known (Khanna,

Linial, Safra 1992), but our proof is novel as it does not rely on

the PCP theorem, while the earlier one does. This ...
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Jonas Holmerin

We prove that Minimum vertex cover on 4-regular hyper-graphs (or

in other words, hitting set where all sets have size exactly 4),

is hard to approximate within 2 - \epsilon.

We also prove that the maximization version, in which we

are allowed to pick ...
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Oded Goldreich

Using known results regarding PCP,

we present simple proofs of the inapproximability

of vertex cover for hypergraphs.

Specifically, we show that

(1) Approximating the size of the minimum vertex cover

in $O(1)$-regular hypergraphs to within a factor of~1.99999 is NP-hard.

(2) Approximating the size ...
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Irit Dinur, Venkatesan Guruswami, Subhash Khot

Given a $k$-uniform hypergraph, the E$k$-Vertex-Cover problem is

to find a minimum subset of vertices that ``hits'' every edge. We

show that for every integer $k \geq 5$, E$k$-Vertex-Cover is

NP-hard to approximate within a factor of $(k-3-\epsilon)$, for

an arbitrarily small constant $\epsilon > 0$.

This almost matches the ... more >>>

Oded Goldreich, Madhu Sudan

Locally testable codes are error-correcting codes that admit

very efficient codeword tests. Specifically, using a constant

number of (random) queries, non-codewords are rejected with

probability proportional to their distance from the code.

Locally testable codes are believed to be the combinatorial

core of PCPs. However, the relation is ...
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Eli Ben-Sasson, Madhu Sudan

We continue the investigation of locally testable codes, i.e.,

error-correcting codes for whom membership of a given word in the

code can be tested probabilistically by examining it in very few

locations. We give two general results on local testability:

First, motivated by the recently proposed notion of robust

probabilistically ...
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Eli Ben-Sasson, Madhu Sudan

We give constructions of PCPs of length n*polylog(n) (with respect

to circuits of size n) that can be verified by making polylog(n)

queries to bits of the proof. These PCPs are not only shorter than

previous ones, but also simpler. Our (only) building blocks are

Reed-Solomon codes and the bivariate ...
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Oded Goldreich

The notion of promise problems was introduced and initially studied

by Even, Selman and Yacobi

(Information and Control, Vol.~61, pages 159-173, 1984).

In this article we survey some of the applications that this

notion has found in the twenty years that elapsed.

These include the notion ...
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Irit Dinur

Let C={c_1,...,c_n} be a set of constraints over a set of

variables. The {\em satisfiability-gap} of C is the smallest

fraction of unsatisfied constraints, ranging over all possible

assignments for the variables.

We prove a new combinatorial amplification lemma that doubles the

satisfiability-gap of a constraint-system, with only a linear ...
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Dana Moshkovitz, Ran Raz

Given a function f:F^m \rightarrow F over a finite

field F, a low degree tester tests its proximity to

an m-variate polynomial of total degree at most d

over F. The tester is usually given access to an oracle

A providing the supposed restrictions of f to

affine subspaces of ...
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Oded Goldreich

We highlight a common theme in four relatively recent works

that establish remarkable results by an iterative approach.

Starting from a trivial construct,

each of these works applies an ingeniously designed

sequence of iterations that yields the desired result,

which is highly non-trivial. Furthermore, in each iteration,

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Charanjit Jutla

Dana Moshkovitz, Ran Raz

We show a construction of a PCP with both sub-constant error and

almost-linear size. Specifically, for some constant alpha in (0,1),

we construct a PCP verifier for checking satisfiability of

Boolean formulas that on input of size n uses log n + O((log

n)^{1-alpha}) random bits to query a constant ...
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Yael Tauman Kalai, Ran Raz

An interactive-PCP (say, for the membership $x \in L$) is a

proof that can be verified by reading only one of its bits, with the

help of a very short interactive-proof.

We show that for membership in some languages $L$, there are

interactive-PCPs that are significantly shorter than the known

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Matthew Andrews, Julia Chuzhoy, Venkatesan Guruswami, Sanjeev Khanna, Kunal Talwar, Lisa Zhang

In the undirected Edge-Disjoint Paths problem with Congestion

(EDPwC), we are given an undirected graph with $V$ nodes, a set of

terminal pairs and an integer $c$. The objective is to route as many

terminal pairs as possible, subject to the constraint that at most

$c$ demands can be routed ...
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Venkatesan Guruswami, Prasad Raghavendra

We study the approximability of the \maxcsp problem over non-boolean domains, more specifically over $\{0,1,\ldots,q-1\}$ for some integer $q$. We obtain a approximation algorithm that achieves a ratio of $C(q) \cdot k/q^k$ for some constant $C(q)$ depending only on $q$. Further, we extend the techniques of Samorodnitsky and Trevisan to ... more >>>

Or Meir

We define a non-uniform model of PCPs of Proximity, and observe that in this model the non-uniform verifiers can always be made very efficient. Specifically, we show that any non-uniform verifier can be modified to run in time that is roughly polynomial in its randomness and query complexity.

more >>>Irit Dinur, Prahladh Harsha

The main result of this paper is a simple, yet generic, composition theorem for low error two-query probabilistically checkable proofs (PCPs). Prior to this work, composition of PCPs was well-understood only in the constant error regime. Existing composition methods in the low error regime were non-modular (i.e., very much tailored ... more >>>

Gábor Kun, Mario Szegedy

The well known dichotomy conjecture of Feder and

Vardi states that for every ﬁnite family Γ of constraints CSP(Γ) is

either polynomially solvable or NP-hard. Bulatov and Jeavons re-

formulated this conjecture in terms of the properties of the algebra

P ol(Γ), where the latter is ...
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Venkatesan Guruswami, Ali Kemal Sinop

We study the maximization version of the fundamental graph coloring problem. Here the goal is to color the vertices of a $k$-colorable graph with $k$ colors so that a maximum fraction of edges are properly colored (i.e., their endpoints receive different colors). A random $k$-coloring properly colors an expected fraction ... more >>>

Gillat Kol, Ran Raz

The Unique Games Conjecture (UGC) is possibly the most important open problem in the research of PCPs and hardness of approximation. The conjecture is a strengthening of the PCP Theorem, predicting the existence of a special type of PCP verifiers: 2-query verifiers that only make unique tests. Moreover, the UGC ... more >>>

Gillat Kol, Ran Raz

We study Locally Testable Codes (LTCs) that can be tested by making two queries to the tested word using an affine test. That is, we consider LTCs over a finite field F, with codeword testers that only use tests of the form $av_i + bv_j = c$, where v is ... more >>>

Irit Dinur, Or Meir

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 ... more >>>

Subhash Khot, Dana Moshkovitz

In this paper, we consider the problem of approximately solving a system of homogeneous linear equations over reals, where each

equation contains at most three variables.

Since the all-zero assignment always satisfies all the equations exactly, we restrict the assignments to be ``non-trivial". Here is

an informal statement of our ...
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Vitaly Feldman, Venkatesan Guruswami, Prasad Raghavendra, Yi Wu

We prove the following strong hardness result for learning: Given a distribution of labeled examples from the hypercube such that there exists a monomial consistent with $(1-\epsilon)$ of the examples, it is $\mathrm{NP}$-hard to find a halfspace that is correct on $(1/2+\epsilon)$ of the examples, for arbitrary constants $\epsilon ... more >>>

Or Meir

The PCP theorem asserts the existence of proofs that can be verified by a verifier that reads only a very small part of the proof. The theorem was originally proved by Arora and Safra (J. ACM 45(1)) and Arora et al. (J. ACM 45(3)) using sophisticated algebraic tools. More than ... more >>>

Dana Moshkovitz

In this paper we put forward a conjecture: an instantiation of the Sliding Scale Conjecture of Bellare, Goldwasser, Lund and Russell to projection games. We refer to this conjecture as the Projection Games Conjecture.

We further suggest the research agenda of establishing new hardness of approximation results based on the ... more >>>

Eli Ben-Sasson, Alessandro Chiesa, Daniel Genkin, Eran Tromer

Probabilistically-Checkable Proofs (PCPs) form the algorithmic core that enables succinct verification of long proofs/computations in many cryptographic constructions, such as succinct arguments and proof-carrying data.

Despite the wonderful asymptotic savings they bring, PCPs are also the infamous computational bottleneck preventing these cryptographic constructions from being used in practice. This reflects ... more >>>

Venkatesan Guruswami, Sushant Sachdeva, Rishi Saket

We study the problem of computing the minimum vertex cover on $k$-uniform $k$-partite hypergraphs when the $k$-partition is given. On bipartite graphs ($k=2$), the minimum vertex cover can be computed in polynomial time. For $k \ge 3$, this problem is known to be NP-hard. For general $k$, the problem was ... more >>>

Eli Ben-Sasson, Yohay Kaplan, Swastik Kopparty, Or Meir, Henning Stichtenoth

The PCP theorem (Arora et. al., J. ACM 45(1,3)) says that every NP-proof can be encoded to another proof, namely, a probabilistically checkable proof (PCP), which can be tested by a verifier that queries only a small part of the PCP. A natural question is how large is the blow-up ... more >>>

Or Meir

The PCP theorem (Arora et. al., J. ACM 45(1,3)) asserts the existence of proofs that can be verified by reading a very small part of the proof. Since the discovery of the theorem, there has been a considerable work on improving the theorem in terms of the length of the ... more >>>

Per Austrin, Venkatesan Guruswami, Johan Håstad

We prove the following hardness result for a natural promise variant of the classical CNF-satisfiability problem: Given a CNF-formula where each clause has width $w$ and the guarantee that there exists an assignment satisfying at least $g = \lceil \frac{w}{2}\rceil -1$ literals in each clause, it is NP-hard to find ... more >>>

Scott Aaronson, Russell Impagliazzo, Dana Moshkovitz

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 ... more >>>

Eli Ben-Sasson, Emanuele Viola

We construct a PCP for NTIME(2$^n$) with constant

soundness, $2^n \poly(n)$ proof length, and $\poly(n)$

queries where the verifier's computation is simple: the

queries are a projection of the input randomness, and the

computation on the prover's answers is a 3CNF. The

previous upper bound for these two computations was

more >>>

Oded Goldreich, Tom Gur, Ilan Komargodski

Locally testable codes (LTCs) are error-correcting codes

that admit very efficient codeword tests. An LTC is said to

be strong if it has a proximity-oblivious tester;

that is, a tester that makes only a constant number of queries

and reject non-codewords with probability that depends solely

on their distance from ...
more >>>

Subhash Khot, Rishi Saket

We show that it is quasi-NP-hard to color $2$-colorable $12$-uniform hypergraphs with $2^{(\log n)^{\Omega(1) }}$ colors where $n$ is the number of vertices. Previously, Guruswami et al. [GHHSV14] showed that it is quasi-NP-hard to color $2$-colorable $8$-uniform hypergraphs with $2^{2^{\Omega(\sqrt{\log \log n})}}$ colors. Their result is obtained by composing a ... more >>>

Subhash Khot, Dana Moshkovitz

We propose a candidate Lasserre integrality gap construction for the Unique Games problem.

Our construction is based on a suggestion in [KM STOC'11] wherein the authors study the complexity of approximately solving a system of linear equations over reals and suggest it as an avenue towards a (positive) resolution ...
more >>>

Irit Dinur, Prahladh Harsha, Guy Kindler

We show that every language in NP has a PCP verifier that tosses $O(\log n)$ random coins, has perfect completeness, and a soundness error of at most $1/poly(n)$, while making at most $O(poly\log\log n)$ queries into a proof over an alphabet of size at most $n^{1/poly\log\log n}$. Previous constructions that ... more >>>

Eli Ben-Sasson, Alessandro Chiesa, Ariel Gabizon, Madars Virza

The seminal result that every language having an interactive proof also has a zero-knowledge interactive proof assumes the existence of one-way functions. Ostrovsky and Wigderson (ISTCS 1993) proved that this assumption is necessary: if one-way functions do not exist, then only languages in BPP have zero-knowledge interactive proofs.

Ben-Or et ... more >>>

Guy Kindler

The first part of this thesis strengthens the low-error PCP

characterization of NP, coming closer to the upper limit of the

conjecture of~\cite{BGLR}.

In the second part we show that a boolean function over

$n$ variables can be tested for the property of depending ...
more >>>

Oded Goldreich, Tom Gur

We initiate a study of ``universal locally testable codes" (universal-LTCs). These codes admit local tests for membership in numerous possible subcodes, allowing for testing properties of the encoded message. More precisely, a universal-LTC $C:\{0,1\}^k \to \{0,1\}^n$ for a family of functions $\mathcal{F} = \{ f_i : \{0,1\}^k \to \{0,1\} \}_{i ... more >>>

Eli Ben-Sasson, iddo Ben-Tov, Ariel Gabizon, Michael Riabzev

A Probabilistically Checkable Proof of Proximity (PCPP) for a linear code $C$, enables to determine very efficiently if a long input $x$, given as an oracle, belongs to $C$ or is far from $C$.

PCPPs are often a central component of constructions of Probabilistically Checkable Proofs (PCP)s [Babai et al. ...
more >>>

Zvika Brakerski, Justin Holmgren, Yael Tauman Kalai

We present an adaptive and non-interactive protocol for verifying arbitrary efficient computations in fixed polynomial time. Our protocol is computationally sound and can be based on any computational PIR scheme, which in turn can be based on standard polynomial-time cryptographic assumptions (e.g. the worst case hardness of polynomial-factor approximation of ... more >>>

Subhash Khot

We present a candidate reduction from the $3$-Lin problem to the $2$-to-$2$ Games problem and present a combinatorial hypothesis about

Grassmann graphs which, if correct, is sufficient to show the soundness of the reduction in

a certain non-standard sense. A reduction that is sound in this non-standard sense

implies that ...
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Irit Dinur

We show that if gap-3SAT has no sub-exponential time algorithms then a weak form of the sliding scale conjecture holds. Namely, for every $\alpha>0$ any algorithm for $n^\alpha$-approximating the value of label cover must run in time at least $n^{\Omega(\exp(1/\alpha))}$, where $n$ is the size of the instance.

Put differently, ... more >>>

Eli Ben-Sasson, iddo Ben-Tov, Ariel Gabizon, Michael Riabzev

Probabilistically Checkable Proofs (PCPs) [Babai et al. FOCS 90; Arora et al. JACM 98] can be used to construct asymptotically efficient cryptographic zero knowledge arguments of membership in any language in NEXP, with minimal communication complexity and computational effort on behalf of both prover and verifier [Babai et al. STOC ... more >>>

Oded Goldreich, Tom Gur

Universal locally testable codes (Universal-LTCs), recently introduced in our companion paper [GG16], are codes that admit local tests for membership in numerous possible subcodes, allowing for testing properties of the encoded message. In this work, we initiate the study of the NP analogue of these codes, wherein the testing procedures ... more >>>

Joshua Brakensiek, Venkatesan Guruswami

The Unique Games Conjecture (UGC) has pinned down the approximability of all constraint satisfaction problems (CSPs), showing that a natural semidefinite programming relaxation offers the optimal worst-case approximation ratio for any CSP. This elegant picture, however, does not apply for CSP instances that are perfectly satisfiable, due to the imperfect ... more >>>

Venkatesan Guruswami, Rishi Saket

A hypergraph is $k$-rainbow colorable if there exists a vertex coloring using $k$ colors such that each hyperedge has all the $k$ colors. Unlike usual hypergraph coloring, rainbow coloring becomes harder as the number of colors increases. This work studies the rainbow colorability of hypergraphs which are guaranteed to be ... more >>>

Subhash Khot, Dor Minzer, Muli Safra

We prove that pseudorandom sets in Grassmann graph have near-perfect expansion as hypothesized in [DKKMS-2]. This completes

the proof of the $2$-to-$2$ Games Conjecture (albeit with imperfect completeness) as proposed in [KMS, DKKMS-1], along with a

contribution from [BKT].

The Grassmann graph $Gr_{global}$ contains induced subgraphs $Gr_{local}$ that are themselves ... more >>>

Irit Dinur, Oded Goldreich, Tom Gur

We show that every set in $\cal P$ is strongly testable under a suitable encoding. By ``strongly testable'' we mean having a (proximity oblivious) tester that makes a constant number of queries and rejects with probability that is proportional to the distance of the tested object from the property. By ... more >>>

Amey Bhangale

We give very short and simple proofs of the following statements: Given a $2$-colorable $4$-uniform hypergraph on $n$ vertices,

(1) It is NP-hard to color it with $\log^\delta n$ colors for some $\delta>0$.

(2) It is $quasi$-NP-hard to color it with $O\left({\log^{1-o(1)} n}\right)$ colors.

In terms of ... more >>>

Subhash Khot, Dor Minzer, Dana Moshkovitz, Muli Safra

This paper studies expansion properties of the (generalized) Johnson Graph. For natural numbers

t < l < k, the nodes of the graph are sets of size l in a universe of size k. Two sets are connected if

their intersection is of size t. The Johnson graph arises often ...
more >>>

Venkatesan Guruswami, Sai Sandeep

A $k$-uniform hypergraph is said to be $r$-rainbow colorable if there is an $r$-coloring of its vertices such that every hyperedge intersects all $r$ color classes. Given as input such a hypergraph, finding a $r$-rainbow coloring of it is NP-hard for all $k \ge 3$ and $r \ge 2$. ... more >>>

Venkatesan Guruswami, Sai Sandeep

The $d$-to-$1$ conjecture of Khot asserts that it is hard to satisfy an $\epsilon$ fraction of constraints of a satisfiable $d$-to-$1$ Label Cover instance, for arbitrarily small $\epsilon > 0$. We prove that the $d$-to-$1$ conjecture for any fixed $d$ implies the hardness of coloring a $4$-colorable graph with $C$ ... more >>>

Noga Ron-Zewi, Ron Rothblum

Interactive oracle proofs (IOPs) are a hybrid between interactive proofs and PCPs. In an IOP the prover is allowed to interact with a verifier (like in an interactive proof) by sending relatively long messages to the verifier, who in turn is only allowed to query a few of the bits ... more >>>

Mark Braverman, Subhash Khot, Dor Minzer

We propose a variant of the $2$-to-$1$ Games Conjecture that we call the Rich $2$-to-$1$ Games Conjecture and show that it is equivalent to the Unique Games Conjecture. We are motivated by two considerations. Firstly, in light of the recent proof of the $2$-to-$1$ Games Conjecture, we hope to understand ... more >>>

Per Austrin, Jonah Brown-Cohen, Johan Håstad

The factor graph of an instance of a constraint satisfaction problem (CSP) is the bipartite graph indicating which variables appear in each constraint. An instance of the CSP is given by the factor graph together with a list of which predicate is applied for each constraint. We establish that many ... more >>>

Dorit Aharonov, Alex Bredariol Grilo

Despite the interest in the complexity class MA, the randomized analog of NP, there is just a couple of known natural (promise-)MA-complete problems, the first due to Bravyi and Terhal (SIAM Journal of Computing 2009) and the second due to Bravyi (Quantum Information and Computation 2015). Surprisingly, both problems are ... more >>>

Amey Bhangale, Prahladh Harsha, Orr Paradise, Avishay Tal

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 ... more >>>

Dana Moshkovitz

We show that NP-hardness of approximating Boolean unique games on small set expanders can be amplified to the full Unique Games Conjecture on small set expanders.

The latter conjecture is known to imply hardness results for problems like Balanced-Separator, Minimum-Linear-Rearrangement and Small-Set-Expansion that are not known under the Unique ...
more >>>

Liron Bronfman, Ron Rothblum

Modern cryptography fundamentally relies on the assumption that the adversary trying to break the scheme is computationally bounded. This assumption lets us construct cryptographic protocols and primitives that are known to be impossible otherwise. In this work we explore the effect of bounding the adversary's power in other information theoretic ... more >>>

Amey Bhangale, Subhash Khot, Dor Minzer

We prove a stability result for general $3$-wise correlations over distributions satisfying mild connectivity properties. More concretely, we show that if $\Sigma,\Gamma$ and $\Phi$ are alphabets of constant size, and $\mu$ is a pairwise connected distribution over $\Sigma\times\Gamma\times\Phi$ with no $(\mathbb{Z},+)$ embeddings in which the probability of each atom is ... more >>>

Dor Minzer, Kai Zhe Zheng

We show that for all $\varepsilon>0$, for sufficiently large prime power $q\in\mathbb{N}$, for all $\delta>0$, it is NP-hard to distinguish whether a $2$-Prover-$1$-Round projection game with alphabet size $q$ has value at least $1-\delta$, or value at most $1/q^{1-\varepsilon}$. This establishes a nearly optimal alphabet-to-soundness tradeoff for $2$-query PCPs ... more >>>