3 research outputs found
Constraints for symmetry breaking in graph representation
Many complex combinatorial problems arising from a range of scientific
applications (such as computer networks, mathematical chemistry and
bioinformatics) involve searching for an undirected graph satisfying a given
property. Since for any possible solution there can be a large number of isomorphic
representations, these problems can quickly become intractable. One
way to mitigate this problem is to eliminate as many isomorphic copies as
possible by breaking symmetry during search - i.e. by introducing constraints
that ensure that at least one representative graph is generated for each equivalence class, but not the entire class. The goal is to generate as few members
of each class as possible - ideally exactly one: the symmetry break is said
to be complete in this case. In this paper we introduce novel, effective and
compact, symmetry breaking constraints for undirected graph search. While
incomplete, these prove highly beneficial in pruning the search for a graph.
We illustrate the application of symmetry breaking in graph representation to
resolve several open instances in extremal graph theory. We also illustrate the application of our approach to graph edge coloring problems which exhibit additional
symmetries due to the fact that the colors of the edges in any solution
can be permuted
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