Almost-Uniform Sampling of Points on High-Dimensional Algebraic Varieties

We consider the problem of uniform sampling of points on an algebraic variety. Specifically, we develop a randomized algorithm that, given a small set of multivariate polynomials over a sufficiently large finite field, produces a common zero of the polynomials almost uniformly at random. The statistical distance between the output distribution of the algorithm and the uniform distribution on the set of common zeros is polynomially small in the field size, and the running time of the algorithm is polynomial in the description of the polynomials and their degrees provided that the number of the polynomials is a constant

Improved inapproximability factors for some Σ^p₂ minimization problems

We give improved inapproximability results for some minimization problems in the second level of the Polynomial-Time Hierarchy. Extending previous work by Umans [Uma99], we show that several variants of DNF minimization are Σ^p₂-hard to approximate to within factors of n^(1/3−ϵ) and ^(n1/2−ϵ) (where the previous results achieved n^(1/4−ϵ)), for arbitrarily small constant ϵ > 0. For one problem shown to be inapproximable to within n^(1/2−ϵ), we give a matching O(n^(1/2))-approximation algorithm, running in randomized polynomial time with access to an NP oracle, which shows that this result is tight assuming the PH doesn't collapse

Balancing Scalability and Uniformity in SAT Witness Generator

Constrained-random simulation is the predominant approach used in the industry for functional verification of complex digital designs. The effectiveness of this approach depends on two key factors: the quality of constraints used to generate test vectors, and the randomness of solutions generated from a given set of constraints. In this paper, we focus on the second problem, and present an algorithm that significantly improves the state-of-the-art of (almost-)uniform generation of solutions of large Boolean constraints. Our algorithm provides strong theoretical guarantees on the uniformity of generated solutions and scales to problems involving hundreds of thousands of variables.Comment: This is a full version of DAC 2014 pape

Distribution-Aware Sampling and Weighted Model Counting for SAT

Given a CNF formula and a weight for each assignment of values to variables, two natural problems are weighted model counting and distribution-aware sampling of satisfying assignments. Both problems have a wide variety of important applications. Due to the inherent complexity of the exact versions of the problems, interest has focused on solving them approximately. Prior work in this area scaled only to small problems in practice, or failed to provide strong theoretical guarantees, or employed a computationally-expensive maximum a posteriori probability (MAP) oracle that assumes prior knowledge of a factored representation of the weight distribution. We present a novel approach that works with a black-box oracle for weights of assignments and requires only an {\NP}-oracle (in practice, a SAT-solver) to solve both the counting and sampling problems. Our approach works under mild assumptions on the distribution of weights of satisfying assignments, provides strong theoretical guarantees, and scales to problems involving several thousand variables. We also show that the assumptions can be significantly relaxed while improving computational efficiency if a factored representation of the weights is known.Comment: This is a full version of AAAI 2014 pape

Uniform and scalable SAT-sampling for configurable systems

Several relevant analyses on configurable software systems remain intractable because they require examining vast and highly-constrained configuration spaces. Those analyses could be addressed through statistical inference, i.e., working with a much more tractable sample that later supports generalizing the results obtained to the entire configuration space. To make this possible, the laws of statistical inference impose an indispensable requirement: each member of the population must be equally likely to be included in the sample, i.e., the sampling process needs to be "uniform". Various SAT-samplers have been developed for generating uniform random samples at a reasonable computational cost. Unfortunately, there is a lack of experimental validation over large configuration models to show whether the samplers indeed produce genuine uniform samples or not. This paper (i) presents a new statistical test to verify to what extent samplers accomplish uniformity and (ii) reports the evaluation of four state-of-the-art samplers: Spur, QuickSampler, Unigen2, and Smarch. According to our experimental results, only Spur satisfies both scalability and uniformity.Ministerio de Ciencia, Innovación y Universidades VITAL-3D DPI2016-77677-PMinisterio de Ciencia, Innovación y Universidades OPHELIA RTI2018-101204-B-C22Comunidad Autónoma de Madrid CAM RoboCity2030 S2013/MIT-2748Agencia Estatal de Investigación TIN2017-90644-RED