3 research outputs found
Computing the maximal canonical form for trees in polynomial time
Known algorithms computing a canonical form for trees in linear time use specialized canonical forms for trees and no canonical forms defined for all graphs. For a graph the maximal canonical form is obtained by relabelling the vertices with in a way that the binary number with bits that is the result of concatenating the rows of the adjacency matrix is maximal. This maximal canonical form is not only defined for all graphs but even plays a special role among the canonical forms for graphs due to some nesting properties allowing orderly algorithms. We give an algorithm to compute the maximal canonical form of a tree
Local Message Passing on Frustrated Systems
Message passing on factor graphs is a powerful framework for probabilistic
inference, which finds important applications in various scientific domains.
The most wide-spread message passing scheme is the sum-product algorithm (SPA)
which gives exact results on trees but often fails on graphs with many small
cycles. We search for an alternative message passing algorithm that works
particularly well on such cyclic graphs. Therefore, we challenge the extrinsic
principle of the SPA, which loses its objective on graphs with cycles. We
further replace the local SPA message update rule at the factor nodes of the
underlying graph with a generic mapping, which is optimized in a data-driven
fashion. These modifications lead to a considerable improvement in performance
while preserving the simplicity of the SPA. We evaluate our method for two
classes of cyclic graphs: the 2x2 fully connected Ising grid and factor graphs
for symbol detection on linear communication channels with inter-symbol
interference. To enable the method for large graphs as they occur in practical
applications, we develop a novel loss function that is inspired by the Bethe
approximation from statistical physics and allows for training in an
unsupervised fashion.Comment: To appear at UAI 202
Connected cubic graphs with the maximum number of perfect matchings
It is proved that for , the number of perfect matchings in a simple
connected cubic graph on vertices is at most , with being
the -th Fibonacci number. The unique extremal graph is characterized as
well. In addition, it is shown that the number of perfect matchings in any
cubic graph equals the expected value of a random variable defined on all
-colorings of edges of . Finally, an improved lower bound on the maximum
number of cycles in a cubic graph is provided.Comment: 20 pages, 19 figure