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Complexity of counting the optimal solutions

2009
*
Theoretical Computer Science
*

In particular,

doi:10.1016/j.tcs.2009.05.025
fatcat:64qcmnafzjdt7pgq2vf4umtgsu
*the*classes #·Π k P*with*k ≥ 1 corresponding*to*all levels*of**the*polynomial hierarchy have thus been studied. ... We prove several important properties*of*these new classes, like closure properties*and**the*relationship*with**the*#·Π k P-classes. ... Acknowledgment: We thank Arnaud Durand*and*Yann Strozecki for their remarks*and*for*the*discussion on*the*proof*of*Theorem 9. ...##
###
POLYNOMIAL 3-SAT REDUCTION OF SUDOKU PUZZLE

2018
*
International Journal of Advanced Research in Computer Science
*

Thus, we have obtained polynomial 3-

doi:10.26483/ijarcs.v9i3.6096
fatcat:p2z34m3thrglxhofkgwmrujkde
*SAT**reduction**of*Sudoku Puzzle (n x n) as well as total*number**of*3-*SAT*clauses*and*new*variables*generated in 3-*SAT**reduction*are 4 [n 4 -2n 2 + m]*and*2 [n 2 {n 2 ... 3-Satisfiability (3-*SAT*)*reduction*has always been remarkable asset in proving*the*NP-Completeness*of*other*problems*. 3-*SAT**problem*is an NP-Complete*problem*used as a starting point*to*prove*the*hardness ... By using equation(1)*and*table 1, we get*the*total*number**of*3-*SAT*clauses (say |C|)*and*new*variables*(say |V|) used in 3-*SAT**reduction**of*Sudoku Puzzle: depicts*the*results*of*our implementation on ...##
###
Positive 1-in-3-SAT admits a non-trivial Kernel
[article]

2020
*
arXiv
*
pre-print

We obtain an upper bound for

arXiv:1808.02821v6
fatcat:5p4olgashbb6ho4xxwwtqrurje
*the*complexity*of*counting, O(2κ r 2^(1-κ) r) for*number**of**variables*r*and*clauses*to**variables*ratio κ. ... Any such instance*of*|V|*variables**and*|C| clauses can be polynomial-time reduced*to*an instance*of*0/1 Integer Programming*with*Equality,*of*size at most 2/3|V|*variables**and*at most |C| clauses. ... Jansen*and*Pieterse are not primarily interested in*reduction**of**the**number**of**variables*, only in*reduction**of**number**of*constraints. They do not tackle*the*associated counting*problem*. ...##
###
Hard satisfiable 3-SAT instances via autocorrelation

2016
*
Journal on Satisfiability, Boolean Modeling and Computation
*

We generated a family

doi:10.3233/sat190111
fatcat:4nhqvae34bcpnnpzvangax7tsi
*of*3-*SAT*instances from*the*D-optimal matrices*problem**with**the*motivation*to*solve interesting cases using*the*power*of**SAT*solvers. ... Our*reduction*techniques is fairly straightforward*and*can be adapted*to*various other*problems*that are defined via autocorrelation. ... During*the*workshop we were able*to*discuss, clarify*and*finalize several key aspects*of*this paper. ...##
###
A Comparison of Local Reduction and Sat-Solver Based Algebraic Cryptanalysis of Jh And Keccak

2012
*
Tatra Mountains Mathematical Publications
*

We also compare

doi:10.2478/v10127-012-0036-6
fatcat:s6teywqgirciho6ixcxg4ke5zy
*the*results based on*the*local*reduction**with**the*attack based on*the*use*of**SAT*-solvers PrecoSAT,*and*CryptoMiniSAT,*respectively*. ... In*the*article we show*the*application*of*this method*to*evaluate*the*strength*of*(reduced versions*of*) two chosen SHA-3 candidates: JH,*and*Keccak,*respectively*. ...*The*authors are grateful*to*an anonymous referee for helpful comments. ...##
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On Moderately Exponential Time for SAT
[chapter]

2010
*
Lecture Notes in Computer Science
*

In this paper, we relate

doi:10.1007/978-3-642-14186-7_27
fatcat:vwu4po4v7jac7n4ifktf3wqctu
*the*question*of*moderately exponential complexity*of**sat**to**the*question*of*moderately exponential complexity*of**problems*defined by existential second-order sentences. ... Can*sat*be solved in "moderately exponential" time, i.e., in time p(|F |) 2 cn for some polynomial p*and*some constant c < 1, where F is a CNF formula*of*size |F | over n*variables*? ... Acknowledgment We are grateful*to**the*anonymous referees for their careful analysis*and*valuable suggestions. We implemented most*of**the*suggestions in*the*final version. ...##
###
Improved SAT-based Reachability Analysis with Observability Don't Cares

2008
*
Journal on Satisfiability, Boolean Modeling and Computation
*

Since ODCs represent

doi:10.3233/sat190050
fatcat:clfuhzj4wbbufmn22xn3rhlmxa
*variables*whose values do not affect*the*outcome*of*a*problem*, it is possible*to*satisfy a*problem**with*fewer assigned*variables*. ... This work proposes*to*use*the*ODCs*to*improve*the*quality*and*performance*of*SATbased reachability analysis frameworks. ...*The*score for each*variable*is calculated in a quick linear time pre-processing step*with**respect**to**the**problem*size. ...##
###
On the Subexponential Time Complexity of CSP
[article]

2013
*
arXiv
*
pre-print

We first establish relations between

arXiv:1304.1996v1
fatcat:6lg37c6elrdbfl24dh3bm2x3ju
*the*subexponential-time complexity*of*CSP*and*that*of*other*problems*, including CNF-*Sat*. ... For several natural CSP parameters, we obtain threshold functions that precisely dictate*the*subexponential-time complexity*of*CSP*with**respect**to**the*parameters under consideration. ... In*the*case*of*CSP*and*CNF-*SAT*, µ*and*µ ′ will be*the**number**of**variables*in*the*instances*of*these*problems*. ...##
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(2+f(n))-SAT and its properties

2004
*
Discrete Applied Mathematics
*

Consider a formula that contains n

doi:10.1016/s0166-218x(03)00194-x
fatcat:rwhorgzlgncw5a2caf7cbiqova
*variables**with**the*form = 2 ∧ 3, where 2 is an instance*of*2-*SAT*containing m2 2-clauses*and*3 is an instance*of*3-*SAT*containing m3 3-clauses. is an instance*of*(2 , ... Most interestingly, we give a candidate, (2 + (log n) k =n 2 )-*SAT*(k ¿ 2), for natural*problems*in NP − NPC − P (denoted as NPI)*with**respect**to*this (2 + f(n))-*SAT*model. ... Acknowledgements*The*authors are grateful*to**the*anonymous referees for their many valuable suggestions*and*constructive criticism that has improved former versions*of*this paper greatly. ...##
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On the Subexponential-Time Complexity of CSP

2015
*
The Journal of Artificial Intelligence Research
*

Not all NP-complete

doi:10.1613/jair.4540
fatcat:kr5w2c53vnd6laa6xbpdnk5ryy
*problems*share*the*same practical hardness*with**respect**to*exact computation. ... We provide tight characterizations*of**the*subexponential-time complexity*of**the*aforementioned*problems**with**respect**to*several natural structural parameters, which allows us*to*draw a detailed landscape ... In*the*case*of*CSP*and*CNF-*SAT*, µ*and*µ will be*the**number**of**variables*in*the*instances*of*these*problems*. ...##
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Planar 3-SAT with a Clause/Variable Cycle

2018
*
Scandinavian Workshop on Algorithm Theory
*

*The*

*problem*remains hard for monotone formulas

*and*instances

*with*exactly three distinct

*variables*per clause. ... In

*the*Planar 3-

*SAT*

*problem*, we are given a 3-

*SAT*formula together

*with*its incidence graph, which is planar,

*and*are asked whether this formula is satisfiable. ... However,

*with*

*the*right perspective on

*the*Linked Planar 3-

*SAT*

*problem*, it will be easy

*to*observe that there are formulas whose incidence graph has a grid minor

*with*a linear

*number*

*of*vertices (

*and*thus ...

##
###
The complexity of counting solutions to Generalised Satisfiability Problems modulo k
[article]

2008
*
arXiv
*
pre-print

In 1979, Valiant introduced

arXiv:0809.1836v1
fatcat:kuigcxlghjdfxcwxtopswq3eny
*the*complexity class parity P,*the**problem**of*counting*the**number**of*solutions*to*NP*problems*modulo two. ... We give a dichotomy theorem for*the*complexity*of*counting*the**number**of*solutions*to*Generalised Satisfiability*Problems*modulo integers. ... We proceed by*reduction*from # p -*SAT*. Given a*SAT*formula F*with*clauses {C 1 , . . . C m }*and**variables*{x 1 , . . . x n }*and*a prime*number*p, we construct a graph as described below. ...##
###
On planar variants of the monotone satisfiability problem with bounded variable appearances
[article]

2016
*
arXiv
*
pre-print

We show NP-completeness for several planar variants

arXiv:1604.05588v1
fatcat:72otsflwxrb35eorur3q573x4a
*of**the*monotone satisfiability*problem**with*bounded*variable*appearances. ... Hence, a planar*and*orthogonal drawing for these graphs can be computed efficiently, which may turn out*to*be useful in*reductions*using these variants as a starting point for proving some decision*problem*... We present hardness proofs for planar*and*monotone variants*of**the*satisfiability*problem**with*bounds on*the**number**of**variable*appearances. ...##
###
Mapping NP-Hard Problems to Restricted Adiabatic Quantum Architectures
[article]

2019
*
arXiv
*
pre-print

Separating

arXiv:1911.00249v2
fatcat:d4c4r3gaqbc7pk3jjlusngnz6u
*the*mapping into two distinct stages, we introduce*problem*-specific*reductions*for both quadratic unconstrained binary optimisation (QUBO)*and*satisfiability (*SAT*)*and*develop*the*subdivision-embedding ... For each*of**the**problem*mapping methods presented, we examine how*the*physical qubit count scales*with**problem*size on architectures*of*varying connectivity. ... For an arbitrary*SAT*instance where*the**number**of**variables*per clause is not constant, Ck may be replaced*with**the*total*number**of**variable*appearances. ...##
###
Strong partial clones and the time complexity of SAT problems

2017
*
Journal of computer and system sciences (Print)
*

In this article we study

doi:10.1016/j.jcss.2016.07.008
fatcat:ydgat5j4f5errokt7zrdpn2yqi
*the*complexity*of**SAT*(S)*with**reductions*increasing*the*amount*of**variables*by a constant (CV-*reductions*) or a constant factor (LV-*reductions*). ... Hence, using clone theory, we obtain a solid understanding*of**the*complexity*of**SAT*(S)*with*CV-*and*LV-*reductions*. ... Acknowledgments We thank Magnus Wahlström for helpful discussions on*the*topic*of*this article,*and*in particular for suggesting*the*proof strategy in Lemma 17. ...
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