12,838 research outputs found
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
Analysis of the computational complexity of solving random satisfiability problems using branch and bound search algorithms
The computational complexity of solving random 3-Satisfiability (3-SAT)
problems is investigated. 3-SAT is a representative example of hard
computational tasks; it consists in knowing whether a set of alpha N randomly
drawn logical constraints involving N Boolean variables can be satisfied
altogether or not. Widely used solving procedures, as the
Davis-Putnam-Loveland-Logeman (DPLL) algorithm, perform a systematic search for
a solution, through a sequence of trials and errors represented by a search
tree. In the present study, we identify, using theory and numerical
experiments, easy (size of the search tree scaling polynomially with N) and
hard (exponential scaling) regimes as a function of the ratio alpha of
constraints per variable. The typical complexity is explicitly calculated in
the different regimes, in very good agreement with numerical simulations. Our
theoretical approach is based on the analysis of the growth of the branches in
the search tree under the operation of DPLL. On each branch, the initial 3-SAT
problem is dynamically turned into a more generic 2+p-SAT problem, where p and
1-p are the fractions of constraints involving three and two variables
respectively. The growth of each branch is monitored by the dynamical evolution
of alpha and p and is represented by a trajectory in the static phase diagram
of the random 2+p-SAT problem. Depending on whether or not the trajectories
cross the boundary between phases, single branches or full trees are generated
by DPLL, resulting in easy or hard resolutions.Comment: 37 RevTeX pages, 15 figures; submitted to Phys.Rev.
Portfolio-based Planning: State of the Art, Common Practice and Open Challenges
In recent years the field of automated planning has significantly
advanced and several powerful domain-independent
planners have been developed. However, none of these systems
clearly outperforms all the others in every known
benchmark domain. This observation motivated the idea of
configuring and exploiting a portfolio of planners to perform
better than any individual planner: some recent planning systems
based on this idea achieved significantly good results in
experimental analysis and International Planning Competitions.
Such results let us suppose that future challenges of the
Automated Planning community will converge on designing
different approaches for combining existing planning algorithms.
This paper reviews existing techniques and provides an exhaustive
guide to portfolio-based planning. In addition, the
paper outlines open issues of existing approaches and highlights
possible future evolution of these techniques
Anytime Computation of Cautious Consequences in Answer Set Programming
Query answering in Answer Set Programming (ASP) is usually solved by
computing (a subset of) the cautious consequences of a logic program. This task
is computationally very hard, and there are programs for which computing
cautious consequences is not viable in reasonable time. However, current ASP
solvers produce the (whole) set of cautious consequences only at the end of
their computation. This paper reports on strategies for computing cautious
consequences, also introducing anytime algorithms able to produce sound answers
during the computation.Comment: To appear in Theory and Practice of Logic Programmin
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