Sanjeev Khanna, Madhu Sudan, David P. Williamson

In this paper we study the approximability of boolean constraint

satisfaction problems. A problem in this class consists of some

collection of ``constraints'' (i.e., functions

$f:\{0,1\}^k \rightarrow \{0,1\}$); an instance of a problem is a set

of constraints applied to specified subsets of $n$ boolean

variables. Schaefer earlier ...
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Sanjeev Khanna, Madhu Sudan, Luca Trevisan

This paper continues the work initiated by Creignou [Cre95] and

Khanna, Sudan and Williamson [KSW96] who classify maximization

problems derived from boolean constraint satisfaction. Here we

study the approximability of {\em minimization} problems derived

thence. A problem in this framework is characterized by a

collection F ...
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Zdenek Dvorák, Daniel Král, Ondrej Pangrác

An instance of a constraint satisfaction problem is $l$-consistent

if any $l$ constraints of it can be simultaneously satisfied.

For a set $\Pi$ of constraint types, $\rho_l(\Pi)$ denotes the largest ratio of constraints which can be satisfied in any $l$-consistent instance composed by constraints from the set $\Pi$. In the ...
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Ondrej Klíma, Pascal Tesson, Denis Thérien

We consider the problem of testing whether a given system of equations

over a fixed finite semigroup S has a solution. For the case where

S is a monoid, we prove that the problem is computable in polynomial

time when S is commutative and is the union of its subgroups

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Víctor Dalmau, Ricard Gavaldà, Pascal Tesson, Denis Thérien

It is well known that coset-generating relations lead to tractable

constraint satisfaction problems. These are precisely the relations closed

under the operation $xy^{-1}z$ where the multiplication is taken in

some finite group. Bulatov et al. have on the other hand shown that

any clone containing the multiplication of some ``block-group'' ...
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Heribert Vollmer, Michael Bauland, Elmar Böhler, Nadia Creignou, Steffen Reith, Henning Schnoor

In this paper we will look at restricted versions of the evaluation problem, the model checking problem, the equivalence problem, and the counting problem for quantified propositional formulas, both with and without bound on the number of quantifier alternations. The restrictions are such that we consider formulas in conjunctive normal-form ... more >>>

Laszlo Egri, Benoit Larose, Pascal Tesson

We introduce symmetric Datalog, a syntactic restriction of linear

Datalog and show that its expressive power is exactly that of

restricted symmetric monotone Krom SNP. The deep result of

Reingold on the complexity of undirected

connectivity suffices to show that symmetric Datalog queries can be

evaluated in logarithmic space. We ...
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Benoit Larose, Pascal Tesson, Pascal Tesson

We present algebraic conditions on constraint languages \Gamma

that ensure the hardness of the constraint satisfaction problem

CSP(\Gamma) for complexity classes L, NL, P, NP and Mod_pL.

These criteria also give non-expressibility results for various

restrictions of Datalog. Furthermore, we show that if

CSP(\Gamma) is not first-order definable then it ...
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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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Jonathan Ullman, Salil Vadhan

Assuming the existence of one-way functions, we show that there is no

polynomial-time, differentially private algorithm $A$ that takes a database

$D\in (\{0,1\}^d)^n$ and outputs a ``synthetic database'' $\hat{D}$ all of whose two-way

marginals are approximately equal to those of $D$. (A two-way marginal is the fraction

of database rows ...
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Yuichi Yoshida

Raghavendra (STOC 2008) gave an elegant and surprising result: if Khot's Unique Games Conjecture (STOC 2002) is true, then for every constraint satisfaction problem (CSP), the best approximation ratio is attained by a certain simple semidefinite programming and a rounding scheme for it.

In this paper, we show that a ...
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Arnab Bhattacharyya, Yuichi Yoshida

Given an instance $\mathcal{I}$ of a CSP, a tester for $\mathcal{I}$ distinguishes assignments satisfying $\mathcal{I}$ from those which are far from any assignment satisfying $\mathcal{I}$. The efficiency of a tester is measured by its query complexity, the number of variable assignments queried by the algorithm. In this paper, we characterize ... more >>>

Subhash Khot, Madhur Tulsiani, Pratik Worah

A boolean predicate $f:\{0,1\}^k\to\{0,1\}$ is said to be {\em somewhat approximation resistant} if for some constant $\tau > \frac{|f^{-1}(1)|}{2^k}$, given a $\tau$-satisfiable instance of the MAX-$k$-CSP$(f)$ problem, it is NP-hard to find an assignment that {\it strictly beats} the naive algorithm that outputs a uniformly random assignment. Let $\tau(f)$ denote ... more >>>

Subhash Khot, Madhur Tulsiani, Pratik Worah

For a predicate $f:\{-1,1\}^k \mapsto \{0,1\}$ with $\rho(f) = \frac{|f^{-1}(1)|}{2^k}$, we call the predicate strongly approximation resistant if given a near-satisfiable instance of CSP$(f)$, it is computationally hard to find an assignment such that the fraction of constraints satisfied is outside the range $[\rho(f)-\Omega(1), \rho(f)+\Omega(1)]$.

We present a characterization of ... more >>>

Subhash Khot, Madhur Tulsiani, Pratik Worah

A predicate $f:\{-1,1\}^k \mapsto \{0,1\}$ with $\rho(f) = \frac{|f^{-1}(1)|}{2^k}$ is called {\it approximation resistant} if given a near-satisfiable instance of CSP$(f)$, it is computationally hard to find an assignment that satisfies at least $\rho(f)+\Omega(1)$ fraction of the constraints.

We present a complete characterization of approximation resistant predicates under the ... more >>>

Gábor Ivanyos, Raghav Kulkarni, Youming Qiao, Miklos Santha, Aarthi Sundaram

In a recent work of Bei, Chen and Zhang (STOC 2013), a trial and error model of computing was introduced, and applied to some constraint satisfaction problems. In this model the input is hidden by an oracle which, for a candidate assignment, reveals some information about a violated constraint if ... more >>>

Amey Bhangale, Swastik Kopparty, Sushant Sachdeva

Given $k$ collections of 2SAT clauses on the same set of variables $V$, can we find one assignment that satisfies a large fraction of clauses from each collection? We consider such simultaneous constraint satisfaction problems, and design the first nontrivial approximation algorithms in this context.

Our main result is that ... more >>>

Venkatesan Guruswami, Euiwoong Lee

A Boolean constraint satisfaction problem (CSP) is called approximation resistant if independently setting variables to $1$ with some probability $\alpha$ achieves the best possible approximation ratio for the fraction of constraints satisfied. We study approximation resistance of a natural subclass of CSPs that we call Symmetric Constraint Satisfaction Problems (SCSPs), ... more >>>

Titus Dose

We study the computational complexity of constraint satisfaction problems that are based on integer expressions and algebraic circuits. On input of a finite set of variables and a finite set of constraints the question is whether the variables can be mapped onto finite subsets of natural numbers (resp., finite intervals ... more >>>

Mrinalkanti Ghosh, Madhur Tulsiani

We study the approximability of constraint satisfaction problems (CSPs) by linear programming (LP) relaxations. We show that for every CSP, the approximation obtained by a basic LP relaxation, is no weaker than the approximation obtained using relaxations given by $\Omega\left(\frac{\log n}{\log \log n}\right)$ levels of the Sherali-Adams hierarchy on instances ... more >>>

Joshua Brakensiek, Venkatesan Guruswami

We give a family of dictatorship tests with perfect completeness and low-soundness for 2-to-2 constraints. The associated 2-to-2 conjecture has been the basis of some previous inapproximability results with perfect completeness. However, evidence towards the conjecture in the form of integrality gaps even against weak semidefinite programs has been elusive. ... more >>>

Alexander Durgin, Brendan Juba

We consider several closely related variants of PAC-learning in which false-positive and false-negative errors are treated differently. In these models we seek to guarantee a given, low rate of false-positive errors and as few false-negative errors as possible given that we meet the false-positive constraint. Bshouty and Burroughs first observed ... more >>>

Chi-Ning Chou, Alexander Golovnev, Madhu Sudan, Santhoshini Velusamy

A Boolean constraint satisfaction problem (CSP), Max-CSP$(f)$, is a maximization problem specified by a constraint $f:\{-1,1\}^k\to\{0,1\}$. An instance of the problem consists of $m$ constraint applications on $n$ Boolean variables, where each constraint application applies the constraint to $k$ literals chosen from the $n$ variables and their negations. The goal ... more >>>

Chi-Ning Chou, Alexander Golovnev, Madhu Sudan, Santhoshini Velusamy

A constraint satisfaction problem (CSP), Max-CSP$({\cal F})$, is specified by a finite set of constraints ${\cal F} \subseteq \{[q]^k \to \{0,1\}\}$ for positive integers $q$ and $k$. An instance of the problem on $n$ variables is given by $m$ applications of constraints from ${\cal F}$ to subsequences of the $n$ ... more >>>

Amey Bhangale, Subhash Khot, Dor Minzer

We consider the $P$-CSP problem for $3$-ary predicates $P$ on satisfiable instances. We show that under certain conditions on $P$ and a $(1,s)$ integrality gap instance of the $P$-CSP problem, it can be translated into a dictatorship vs. quasirandomness test with perfect completeness and soundness $s+\varepsilon$, for every constant $\varepsilon>0$. ... more >>>

Gillat Kol, Dmitry Paramonov, Raghuvansh Saxena, Huacheng Yu

We study boolean constraint satisfaction problems (CSPs) $\mathrm{Max}\text{-}\mathrm{CSP}^f_n$ for all predicates $f: \{ 0, 1 \} ^k \to \{ 0, 1 \}$. In these problems, given an integer $v$ and a list of constraints over $n$ boolean variables, each obtained by applying $f$ to a sequence of literals, we wish ... more >>>

Amey Bhangale, Subhash Khot, Dor Minzer

In this paper we study functions on the Boolean hypercube that have the property that after applying certain random restrictions, the restricted function is correlated to a linear function with non-negligible probability. If the given function is correlated with a linear function then this property clearly holds. Furthermore, the property ... more >>>

Amey Bhangale, Subhash Khot, Dor Minzer

Let $\Sigma$ be an alphabet and $\mu$ be a distribution on $\Sigma^k$ for some $k \geq 2$. Let $\alpha > 0$ be the minimum probability of a tuple in the support of $\mu$ (denoted by $supp(\mu)$). Here, the support of $\mu$ is the set of all tuples in $\Sigma^k$ that ... more >>>