1,803 research outputs found
An Overview of Backtrack Search Satisfiability Algorithms
Propositional Satisfiability (SAT) is often used as the underlying model for a significan
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
Using Restarts in Constraint Programming over Finite Domains - An Experimental Evaluation
The use of restart techniques in complete Satisfiability (SAT) algorithms has made
solving hard real world instances possible. Without restarts such algorithms could not
solve those instances, in practice. State of the art algorithms for SAT use restart
techniques, conflict clause recording (nogoods), heuristics based on activity variable in
conflict clauses, among others. Algorithms for SAT and Constraint problems share many
techniques; however, the use of restart techniques in constraint programming with finite
domains (CP(FD)) is not widely used as it is in SAT. We believe that the use of restarts in
CP(FD) algorithms could also be the key to efficiently solve hard combinatorial
problems.
In this PhD thesis we study restarts and associated techniques in CP(FD) solvers. In
particular, we propose to including in a CP(FD) solver restarts, nogoods and heuristics
based in nogoods as this should improve search algorithms, and, consequently, efficiently
solve hard combinatorial problems.
We thus intend to: a) implement restart techniques (successfully used in SAT) to
solve constraint problems with finite domains; b) implement nogoods (learning) and
heuristics based on nogoods, already in use in SAT and associated with restarts; and c)
evaluate the use of restarts and the interplay with the other implemented techniques.
We have conducted the study in the context of domain splitting backtrack search
algorithms with restarts. We have defined domain splitting nogoods that are extracted
from the last branch of the search algorithm before the restart. And, inspired by SAT
solvers, we were able to use information within those nogoods to successfully help the
variable selection heuristics. A frequent restart strategy is also necessary, since our
approach learns from restarts
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