1,260 research outputs found
Learning to Reason: Leveraging Neural Networks for Approximate DNF Counting
Weighted model counting (WMC) has emerged as a prevalent approach for
probabilistic inference. In its most general form, WMC is #P-hard. Weighted DNF
counting (weighted #DNF) is a special case, where approximations with
probabilistic guarantees are obtained in O(nm), where n denotes the number of
variables, and m the number of clauses of the input DNF, but this is not
scalable in practice. In this paper, we propose a neural model counting
approach for weighted #DNF that combines approximate model counting with deep
learning, and accurately approximates model counts in linear time when width is
bounded. We conduct experiments to validate our method, and show that our model
learns and generalizes very well to large-scale #DNF instances.Comment: To appear in Proceedings of the Thirty-Fourth AAAI Conference on
Artificial Intelligence (AAAI-20). Code and data available at:
https://github.com/ralphabb/NeuralDNF
Closing the Gap Between Short and Long XORs for Model Counting
Many recent algorithms for approximate model counting are based on a
reduction to combinatorial searches over random subsets of the space defined by
parity or XOR constraints. Long parity constraints (involving many variables)
provide strong theoretical guarantees but are computationally difficult. Short
parity constraints are easier to solve but have weaker statistical properties.
It is currently not known how long these parity constraints need to be. We
close the gap by providing matching necessary and sufficient conditions on the
required asymptotic length of the parity constraints. Further, we provide a new
family of lower bounds and the first non-trivial upper bounds on the model
count that are valid for arbitrarily short XORs. We empirically demonstrate the
effectiveness of these bounds on model counting benchmarks and in a
Satisfiability Modulo Theory (SMT) application motivated by the analysis of
contingency tables in statistics.Comment: The 30th Association for the Advancement of Artificial Intelligence
(AAAI-16) Conferenc
Efficient Triangle Counting in Large Graphs via Degree-based Vertex Partitioning
The number of triangles is a computationally expensive graph statistic which
is frequently used in complex network analysis (e.g., transitivity ratio), in
various random graph models (e.g., exponential random graph model) and in
important real world applications such as spam detection, uncovering of the
hidden thematic structure of the Web and link recommendation. Counting
triangles in graphs with millions and billions of edges requires algorithms
which run fast, use small amount of space, provide accurate estimates of the
number of triangles and preferably are parallelizable.
In this paper we present an efficient triangle counting algorithm which can
be adapted to the semistreaming model. The key idea of our algorithm is to
combine the sampling algorithm of Tsourakakis et al. and the partitioning of
the set of vertices into a high degree and a low degree subset respectively as
in the Alon, Yuster and Zwick work treating each set appropriately. We obtain a
running time
and an approximation (multiplicative error), where is the number
of vertices, the number of edges and the maximum number of
triangles an edge is contained.
Furthermore, we show how this algorithm can be adapted to the semistreaming
model with space usage and a constant number of passes (three) over the graph
stream. We apply our methods in various networks with several millions of edges
and we obtain excellent results. Finally, we propose a random projection based
method for triangle counting and provide a sufficient condition to obtain an
estimate with low variance.Comment: 1) 12 pages 2) To appear in the 7th Workshop on Algorithms and Models
for the Web Graph (WAW 2010
Probabilistic Inference Modulo Theories
We present SGDPLL(T), an algorithm that solves (among many other problems)
probabilistic inference modulo theories, that is, inference problems over
probabilistic models defined via a logic theory provided as a parameter
(currently, propositional, equalities on discrete sorts, and inequalities, more
specifically difference arithmetic, on bounded integers). While many solutions
to probabilistic inference over logic representations have been proposed,
SGDPLL(T) is simultaneously (1) lifted, (2) exact and (3) modulo theories, that
is, parameterized by a background logic theory. This offers a foundation for
extending it to rich logic languages such as data structures and relational
data. By lifted, we mean algorithms with constant complexity in the domain size
(the number of values that variables can take). We also detail a solver for
summations with difference arithmetic and show experimental results from a
scenario in which SGDPLL(T) is much faster than a state-of-the-art
probabilistic solver.Comment: Submitted to StarAI-16 workshop as closely revised version of
IJCAI-16 pape
Approximate weighted model integration on DNF structures
Weighted model counting consists of computing the weighted sum of all satisfying assignments of a propositional formula. Weighted model counting is well-known to be #P-hard for exact solving, but admits a fully polynomial randomized approximation scheme when restricted to DNF structures. In this work, we study weighted model integration, a generalization of weighted model counting which involves real variables in addition to propositional variables, and pose the following question: Does weighted model integration on DNF structures admit a fully polynomial randomized approximation scheme? Building on classical results from approximate weighted model counting and approximate volume computation, we show that weighted model integration on DNF structures can indeed be approximated for a class of weight functions. Our approximation algorithm is based on three subroutines, each of which can be a weak (i.e., approximate), or a strong (i.e., exact) oracle, and in all cases, comes along with accuracy guarantees. We experimentally verify our approach over randomly generated DNF instances of varying sizes, and show that our algorithm scales to large problem instances, involving up to 1K variables, which are currently out of reach for existing, general-purpose weighted model integration solvers
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