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

Nadia Creignou, Herve Daude, Olivier Dubois
2001 arXiv   pre-print
In this paper we study 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.  ... 
arXiv:cs/0106001v1 fatcat:pxk2l3wlmncdpokdqsr3qi3ffe

Phase Transition Behavior of Cardinality and XOR Constraints [article]

Yash Pote, Saurabh Joshi, Kuldeep S. Meel
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/.  ... 
arXiv:1910.09755v1 fatcat:swv6o7hjxnbjjbxztimz4xl5ya

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

Wolfgang Barthel, Alexander K. Hartmann, Martin Weigt
2003 Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics  
We give numerical and 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  ... 
doi:10.1103/physreve.67.066104 pmid:16241301 fatcat:pr4wbesanve5dgk35emkruxuna

Smooth and sharp thresholds for random{k}-XOR-CNF satisfiability

Nadia Creignou, Hervé Daudé
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) .  ... 
doi:10.1051/ita:2003014 fatcat:klstufgcvvfk7pl22umxz56cce

Approximating satisfiability transition by suppressing fluctuations [article]

S. Knysh, V.N. Smelyanskiy, R.D. Morris
2004 arXiv   pre-print
We illustrate 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.  ... 
arXiv:cond-mat/0403416v1 fatcat:nikzbdq3kjac5hfw6vmulhpgpy

Algorithms and Lower Bounds for De Morgan Formulas of Low-Communication Leaf Gates

Valentine Kabanets, Sajin Koroth, Zhenjian Lu, Dimitrios Myrisiotis, Igor C. Oliveira, Shubhangi Saraf
2020 Computational Complexity Conference  
Let R^(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)), FORMULAXOR For the case of FORMULAXOR, we get a PRG with better seed length.  ... 
doi:10.4230/lipics.ccc.2020.15 dblp:conf/coco/KabanetsKLMO20 fatcat:jzht3ie3ezg7di7goqaak6z32e

Algorithms and Lower Bounds for de Morgan Formulas of Low-Communication Leaf Gates [article]

Valentine Kabanets, Sajin Koroth, Zhenjian Lu, Dimitrios Myrisiotis, Igor Oliveira
2020 arXiv   pre-print
Let R^(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.  ... 
arXiv:2002.08533v1 fatcat:qfiearqugbaklfftdb2ku7flsa

Approximate Counting of Minimal Unsatisfiable Subsets [chapter]

Jaroslav Bendík, Kuldeep S. Meel
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 .  ... 
doi:10.1007/978-3-030-53288-8_21 fatcat:qjocu3fjj5ey7kijkukanzdrhm

Complexity Theoretic Limitations on Learning Halfspaces [article]

Amit Daniely
2016 arXiv   pre-print
Using 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  ... 
arXiv:1505.05800v2 fatcat:udprem4j7zdvvksswzzeowmwku

Optimal Approximation Algorithms for Reoptimization of Constraint Satisfaction Problems

Victor Alex Mikhailyuk
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 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).  ... 
doi:10.4236/ajor.2013.32025 fatcat:5xfdwbnbvrcv7ohnnwnwos6qoa

Bit-Vector Model Counting using Statistical Estimation [article]

Seonmo Kim, Stephen McCamant
2017 arXiv   pre-print
Adding 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.  ... 
arXiv:1712.07770v1 fatcat:lalr6qj5ynfotbamln5oa25bcu

Bit-Vector Model Counting Using Statistical Estimation [chapter]

Seonmo Kim, Stephen McCamant
2018 Lecture Notes in Computer Science  
Adding 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.  ... 
doi:10.1007/978-3-319-89960-2_8 fatcat:qlaofquayrhbvb6d5hmlzpjrvy

Random 3CNF formulas elude the Lovasz theta function [article]

Uriel Feige, Eran Ofek
2006 arXiv   pre-print
We show that 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.).  ... 
arXiv:cs/0603084v1 fatcat:ttytzigscnbtbpo72udxq7g5mi

Tinted, Detached, and Lazy CNF-XOR Solving and Its Applications to Counting and Sampling [chapter]

Mate Soos, Stephan Gocht, Kuldeep S. Meel
2020 Lecture Notes in Computer Science  
Given a Boolean 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).  ... 
doi:10.1007/978-3-030-53288-8_22 fatcat:5hkp4g2yx5d2pgpzekcym44nsi

Certifying solution geometry in random CSPs: counts, clusters and balance [article]

Jun-Ting Hsieh, Sidhanth Mohanty, Jeff Xu
2021 arXiv   pre-print
An active topic in 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.  ... 
arXiv:2106.12710v1 fatcat:54qtuux4yrcm5kenh5nbvstyfa
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