1,770 research outputs found
On variable-weighted exact satisfiability problems
We show that the NP-hard optimization problems minimum and maximum weight exact satisfiability (XSAT) for a CNF formula C over n propositional variables equipped with arbitrary real-valued weights can be solved in O(|C|2^{0.2441n}) time. To the best of our knowledge, the algorithms presented here are the first handling weighted XSAT optimization versions in non-trivial worst case time. We also investigate the corresponding weighted counting problems, namely we show that the number of all minimum, resp. maximum, weight exact satisfiability solutions of an arbitrarily weighted formula can be determined in O(n^2cdot |C|+2^{0.40567n}) time. In recent years only the unweighted counterparts of these problems have been studied cite{dahl,dahl2,porschen}
On variable-weighted exact satisfiability problems
We show that the NP-hard optimization problems minimum and maximum weight exact satisfiability (XSAT) for a CNF formula C over n propositional variables equipped with arbitrary real-valued weights can be solved in O(|C|2^{0.2441n}) time. To the best of our knowledge, the algorithms presented here are the first handling weighted XSAT optimization versions in non-trivial worst case time. We also investigate the corresponding weighted counting problems, namely we show that the number of all minimum, resp. maximum, weight exact satisfiability solutions of an arbitrarily weighted formula can be determined in O(n^2cdot |C|+2^{0.40567n}) time. In recent years only the unweighted counterparts of these problems have been studied cite{dahl,dahl2,porschen}
Computational Complexity for Physicists
These lecture notes are an informal introduction to the theory of
computational complexity and its links to quantum computing and statistical
mechanics.Comment: references updated, reprint available from
http://itp.nat.uni-magdeburg.de/~mertens/papers/complexity.shtm
Bit-Vector Model Counting using Statistical Estimation
Approximate model counting for bit-vector SMT formulas (generalizing \#SAT)
has many applications such as probabilistic inference and quantitative
information-flow security, but it is computationally difficult. 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 results. 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 as possible. We implement this approach, with an approximate
probability model, as a wrapper around an off-the-shelf SMT solver or SAT
solver. Experimental results show that the implementation is faster than the
most similar previous approaches which used simpler refinement strategies. The
technique also lets us model count formulas over floating-point constraints,
which we demonstrate with an application to a vulnerability in differential
privacy mechanisms
Answer Set Solving with Bounded Treewidth Revisited
Parameterized algorithms are a way to solve hard problems more efficiently,
given that a specific parameter of the input is small. In this paper, we apply
this idea to the field of answer set programming (ASP). To this end, we propose
two kinds of graph representations of programs to exploit their treewidth as a
parameter. Treewidth roughly measures to which extent the internal structure of
a program resembles a tree. Our main contribution is the design of
parameterized dynamic programming algorithms, which run in linear time if the
treewidth and weights of the given program are bounded. Compared to previous
work, our algorithms handle the full syntax of ASP. Finally, we report on an
empirical evaluation that shows good runtime behaviour for benchmark instances
of low treewidth, especially for counting answer sets.Comment: This paper extends and updates a paper that has been presented on the
workshop TAASP'16 (arXiv:1612.07601). We provide a higher detail level, full
proofs and more example
On the van der Waerden numbers w(2;3,t)
We present results and conjectures on the van der Waerden numbers w(2;3,t)
and on the new palindromic van der Waerden numbers pdw(2;3,t). We have computed
the new number w(2;3,19) = 349, and we provide lower bounds for 20 <= t <= 39,
where for t <= 30 we conjecture these lower bounds to be exact. The lower
bounds for 24 <= t <= 30 refute the conjecture that w(2;3,t) <= t^2, and we
present an improved conjecture. We also investigate regularities in the good
partitions (certificates) to better understand the lower bounds.
Motivated by such reglarities, we introduce *palindromic van der Waerden
numbers* pdw(k; t_0,...,t_{k-1}), defined as ordinary van der Waerden numbers
w(k; t_0,...,t_{k-1}), however only allowing palindromic solutions (good
partitions), defined as reading the same from both ends. Different from the
situation for ordinary van der Waerden numbers, these "numbers" need actually
to be pairs of numbers. We compute pdw(2;3,t) for 3 <= t <= 27, and we provide
lower bounds, which we conjecture to be exact, for t <= 35.
All computations are based on SAT solving, and we discuss the various
relations between SAT solving and Ramsey theory. Especially we introduce a
novel (open-source) SAT solver, the tawSolver, which performs best on the SAT
instances studied here, and which is actually the original DLL-solver, but with
an efficient implementation and a modern heuristic typical for look-ahead
solvers (applying the theory developed in the SAT handbook article of the
second author).Comment: Second version 25 pages, updates of numerical data, improved
formulations, and extended discussions on SAT. Third version 42 pages, with
SAT solver data (especially for new SAT solver) and improved representation.
Fourth version 47 pages, with updates and added explanation
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