13 research outputs found
Twenty-Five Comparators is Optimal when Sorting Nine Inputs (and Twenty-Nine for Ten)
This paper describes a computer-assisted non-existence proof of nine-input
sorting networks consisting of 24 comparators, hence showing that the
25-comparator sorting network found by Floyd in 1964 is optimal. As a
corollary, we obtain that the 29-comparator network found by Waksman in 1969 is
optimal when sorting ten inputs.
This closes the two smallest open instances of the optimal size sorting
network problem, which have been open since the results of Floyd and Knuth from
1966 proving optimality for sorting networks of up to eight inputs.
The proof involves a combination of two methodologies: one based on
exploiting the abundance of symmetries in sorting networks, and the other,
based on an encoding of the problem to that of satisfiability of propositional
logic. We illustrate that, while each of these can single handed solve smaller
instances of the problem, it is their combination which leads to an efficient
solution for nine inputs.Comment: 18 page
Solving Graph Coloring Problems with Abstraction and Symmetry
This paper introduces a general methodology, based on abstraction and
symmetry, that applies to solve hard graph edge-coloring problems and
demonstrates its use to provide further evidence that the Ramsey number
. The number is often presented as the unknown Ramsey
number with the best chances of being found "soon". Yet, its precise value has
remained unknown for more than 50 years. We illustrate our approach by showing
that: (1) there are precisely 78{,}892 Ramsey colorings; and (2)
if there exists a Ramsey coloring then it is (13,8,8) regular.
Specifically each node has 13 edges in the first color, 8 in the second, and 8
in the third. We conjecture that these two results will help provide a proof
that no Ramsey coloring exists implying that
A parametric approach for smaller and better encodings of cardinality constraints
Adequate encodings for high-level constraints are a key ingredient for the application of SAT technology. In particular, cardinality constraints state that at most (at least, or exactly) k out of n propositional variables can be true. They are crucial in many applications. Although sophisticated encodings for cardinality constraints exist, it is well known that for small n and k straightforward encodings without auxiliary variables sometimes behave better, and that the choice of the right trade-off between minimizing either the number of variables or the number of clauses is highly application-dependent. Here we build upon previous work on Cardinality Networks to get the best of several worlds: we develop an arc-consistent encoding that, by recursively decomposing the constraint into smaller ones, allows one to decide which encoding to apply to each sub-constraint. This process minimizes a function λ·num- vars + num-clauses, where λ is a parameter that can be tuned by the user. Our careful experimental evaluation shows that (e.g., for λ = 5) this new technique produces much smaller encodings in variables and clauses, and indeed strongly improves SAT solvers' performance.Postprint (author’s final draft