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Approximating the satisfiability threshold for random k-XOR-formulas
[article]

2001
*
arXiv
*
pre-print

In this paper we study

arXiv:cs/0106001v1
fatcat:pxk2l3wlmncdpokdqsr3qi3ffe
*random*linear systems with*k*variables per equation over*the*finite field GF(2), or equivalently*k*-*XOR*-CNF*formulas*. ... In a previous paper Creignou and Daudé proved that*the*phase transition*for**the*consistency (*satisfiability*) of such systems (*formulas*) exhibits a sharp*threshold*. ... In a first step we have made precise*the*link between*the**k*-*XOR*-SAT's phase transition and*the*rank of sparse*random*Boolean matrices. ...##
###
Phase Transition Behavior of Cardinality and XOR Constraints
[article]

2019
*
arXiv
*
pre-print

*The*runtime behavior of

*random*CARD-

*XOR*

*formulas*is unexplored in prior work. In this paper, we present

*the*first rigorous empirical study to characterize

*the*runtime behavior of 1-CARD-

*XOR*

*formulas*. ...

*The*problem of determining

*the*

*satisfiability*of CARD-

*XOR*

*formulas*is a fundamental problem with a wide variety of applications ranging from discrete integration in

*the*field of artificial intelligence ...

*The*computational work

*for*this article was performed on resources of

*the*National Supercomputing Centre, Singapore. https://www.nscc.sg/. ...

##
###
Solving satisfiability problems by fluctuations: The dynamics of stochastic local search algorithms

2003
*
Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
*

We give numerical and

doi:10.1103/physreve.67.066104
pmid:16241301
fatcat:pr4wbesanve5dgk35emkruxuna
*approximate*analytical descriptions of*the*dynamics of such algorithms applied to*random**satisfiability*problems. ... We find two different dynamical regimes, depending on*the*number of constraints per variable:*For*low constraintness,*the*problems are solved efficiently, i.e. in linear time. ... Zecchina*for*helpful discussions. We also thank R. Monasson and G. Semerjian*for*communicating their results [23] prior to publication. WB and AKH obtained financial support ...##
###
Smooth and sharp thresholds for random{k}-XOR-CNF satisfiability

2003
*
RAIRO - Theoretical Informatics and Applications
*

*For*

*k*≥ 3 we show

*the*existence of a sharp

*threshold*

*for*

*the*

*satisfiability*of a

*random*

*k*-

*XOR*-CNF

*formula*, whereas there are smooth

*thresholds*

*for*

*k*= 1 and

*k*= 2. ...

*The*aim of this paper is to study

*the*

*threshold*behavior

*for*

*the*

*satisfiability*property of a

*random*

*k*-

*XOR*-CNF

*formula*or equivalently

*for*

*the*consistency of a

*random*Boolean linear system with

*k*variables ... We will denote by SAT (

*k*) n (p)

*the*probability that

*the*

*random*

*formula*S

*k*(n, p) is

*satisfiable*: SAT (

*k*) n (p) = µ p (

*k*-

*XOR*-SAT) . ...

##
###
Approximating satisfiability transition by suppressing fluctuations
[article]

2004
*
arXiv
*
pre-print

We illustrate

arXiv:cond-mat/0403416v1
fatcat:nikzbdq3kjac5hfw6vmulhpgpy
*the*method*for*three sample problems:*K*-*XOR*-SAT,*K*-SAT and positive 1-in-*K*-SAT. ... Determining*the*location of*satisfiability**threshold*α_c=M/N*for*a number of difficult combinatorial problems is a major open problem in*the*theory of*random*graphs. ... That*the*annealing*approximation**for**the*simplified*formula*fails to predict*the*correct transition suggests that a large number of solutions remains up to*the**satisfiability**threshold*. ...##
###
Algorithms and Lower Bounds for De Morgan Formulas of Low-Communication Leaf Gates

2020
*
Computational Complexity Conference
*

Let R^(

doi:10.4230/lipics.ccc.2020.15
dblp:conf/coco/KabanetsKLMO20
fatcat:jzht3ie3ezg7di7goqaak6z32e
*k*)(𝒢) be*the*maximum*k*-party number-on-forehead*randomized*communication complexity of a function in 𝒢. ... As a corollary, we get an average-case lower bound*for*𝖦𝖨𝖯^k_n against*𝖥𝖮𝖱𝖬𝖴𝖫𝖠*[n^{1.99}]∘𝖯𝖳𝖥^{*k*-1}, i.e., sub-quadratic-size de Morgan*formulas*with degree-(*k*-1) PTF (polynomial*threshold*...*k*• log(*k*) • log(m/δ)),*FORMULA*•*XOR**For**the*case of*FORMULA*•*XOR*, we get a PRG with better seed length. ...##
###
Algorithms and Lower Bounds for de Morgan Formulas of Low-Communication Leaf Gates
[article]

2020
*
arXiv
*
pre-print

Let R^(

arXiv:2002.08533v1
fatcat:qfiearqugbaklfftdb2ku7flsa
*k*)(G) be*the*maximum*k*-party NOF*randomized*communication complexity of G. ... In particular, this implies a nontrivial #SAT algorithm*for**FORMULA*[n^1.99]∘ LTF. (4)*The*Minimum Circuit Size Problem is not in*FORMULA*[n^1.99]∘*XOR*. ... Acknowledgements We would like to thank Rocco Servedio*for*bringing to our attention*the*work by Kalai, Mansour, and Verbin [KMV08] , which is a central ingredient in*the*proof of Theorem 7. ...##
###
Approximate Counting of Minimal Unsatisfiable Subsets
[chapter]

2020
*
Lecture Notes in Computer Science
*

*The*current best approach

*for*MUS counting is to employ a MUS enumeration algorithm, which often does not scale

*for*

*the*cases with a reasonably large number of MUSes. ... Motivated by

*the*success of hashing-based techniques in

*the*context of model counting, we design

*the*first

*approximate*MUS counting procedure with (ε, δ) guarantees, called AMUSIC. ... By To choose a hash function uniformly at

*random*from H

*xor*(n, m), we randomly and independently choose

*the*values of a i,

*k*. ...

##
###
Complexity Theoretic Limitations on Learning Halfspaces
[article]

2016
*
arXiv
*
pre-print

Using

arXiv:1505.05800v2
fatcat:udprem4j7zdvvksswzzeowmwku
*the*recently developed method of*the*author, Linial and Shalev-Shwartz we prove hardness of learning results under a natural assumption on*the*complexity of refuting*random**K*-*XOR**formulas*. ... Under a stronger version of*the*assumption (where*K*can be poly-logarithmic in n), we can take η = 2^-^1-ν(n)*for*arbitrarily small ν>0. ...*The*author thanks Uriel Feige, Vitaly Feldman, Nati Linial and Shai Shalev-Shwartz*for*valuable discussions. Many ...##
###
Optimal Approximation Algorithms for Reoptimization of Constraint Satisfaction Problems

2013
*
American Journal of Operations Research
*

In particular, Hastad [6] [7] [8] showed that Max-E3-Lin-2 and Max 3-Sat are NP-hard to

doi:10.4236/ajor.2013.32025
fatcat:5xfdwbnbvrcv7ohnnwnwos6qoa
*approximate*G*K*...*For*this problem there is*the*name of inapproximability or*the*hardness of*approximation*. ... basis of UGC. polynomial ( d P*the**threshold*"*random*"*approximation*ratio of P). ...##
###
Bit-Vector Model Counting using Statistical Estimation
[article]

2017
*
arXiv
*
pre-print

Adding

arXiv:1712.07770v1
fatcat:lalr6qj5ynfotbamln5oa25bcu
*random*parity constraints (*XOR*streamlining) and then checking*satisfiability*is an effective*approximation*technique, but it requires a prior hypothesis about*the*model count to produce useful ... We propose an approach inspired by statistical estimation to continually refine a probabilistic estimate of*the*model count*for*a*formula*, so that each*XOR*-streamlined query yields as much information ... Any opinions, findings, and conclusions or recommendations expressed in this material are those of*the*authors and do not necessarily reflect*the*views of*the*National Science Foundation. ...##
###
Bit-Vector Model Counting Using Statistical Estimation
[chapter]

2018
*
Lecture Notes in Computer Science
*

Adding

doi:10.1007/978-3-319-89960-2_8
fatcat:qlaofquayrhbvb6d5hmlzpjrvy
*random*parity constraints (*XOR*streamlining) and then checking*satisfiability*is an effective*approximation*technique, but it requires a prior hypothesis about*the*model count to produce useful ... We propose an approach inspired by statistical estimation to continually refine a probabilistic estimate of*the*model count*for*a*formula*, so that each*XOR*-streamlined query yields as much information ... Suppose we have a*formula*f with a known influence log 2 N , and add*k**XOR**random*constraints to*the**formula*. ...##
###
Random 3CNF formulas elude the Lovasz theta function
[article]

2006
*
arXiv
*
pre-print

We show that

arXiv:cs/0603084v1
fatcat:ttytzigscnbtbpo72udxq7g5mi
*for**random**formulas*with m < n^3/2 -o(1) clauses,*the*above approach fails, i.e.*the*ϑ function is likely to return a value of m. ... It is an open question whether there is an efficient refutation algorithm that*for*most such*formulas*proves that they are not*satisfiable*. ... Acknowledgements This work was supported in part by a grant from*the*German-Israeli Foundation*for*Scientific Research and Development (G.I.F.). ...##
###
Tinted, Detached, and Lazy CNF-XOR Solving and Its Applications to Counting and Sampling
[chapter]

2020
*
Lecture Notes in Computer Science
*

Given a Boolean

doi:10.1007/978-3-030-53288-8_22
fatcat:5hkp4g2yx5d2pgpzekcym44nsi
*formula*,*the*problem of counting seeks to estimate*the*number of solutions of F while*the*problem of uniform sampling seeks to sample solutions uniformly at*random*. ...*The*past few years have witnessed*the*rise of hashing-based approaches that use XORbased hashing and employ SAT solvers to solve*the*resulting CNF*formulas*conjuncted with*XOR*constraints. ... Similarly, when invoking BoundedSAT*for*i =*k*after determining that*for*i =*k*− 1, Cnt F,*k*−1 ≥ thresh, we first check how many solutions of F ∧(h*k*−1 ) −1 (0)*satisfy*F ∧ (h*k*) −1 (0). ...##
###
Certifying solution geometry in random CSPs: counts, clusters and balance
[article]

2021
*
arXiv
*
pre-print

An active topic in

arXiv:2106.12710v1
fatcat:54qtuux4yrcm5kenh5nbvstyfa
*the*study of*random*constraint satisfaction problems (CSPs) is*the*geometry of*the*space of*satisfying*or almost*satisfying*assignments as*the*function of*the*density,*for*which a precise ... Inspired by this,*the*starting point*for*our work is*the*following question: what does*the*solution space*for*a*random*CSP look like to an efficient algorithm? ... We would also like to thank Tselil Schramm*for*enlightening discussions on refuting*random*CSPs. ...
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