3,907 research outputs found
Hypergraph Isomorphism for Groups with Restricted Composition Factors
We consider the isomorphism problem for hypergraphs taking as input two hypergraphs over the same set of vertices V and a permutation group ? over domain V, and asking whether there is a permutation ? ? ? that proves the two hypergraphs to be isomorphic. We show that for input groups, all of whose composition factors are isomorphic to a subgroup of the symmetric group on d points, this problem can be solved in time (n+m)^O((log d)^c) for some absolute constant c where n denotes the number of vertices and m the number of hyperedges. In particular, this gives the currently fastest isomorphism test for hypergraphs in general. The previous best algorithm for the above problem due to Schweitzer and Wiebking (STOC 2019) runs in time n^O(d)m^O(1).
As an application of this result, we obtain, for example, an algorithm testing isomorphism of graphs excluding K_{3,h} as a minor in time n^O((log h)^c). In particular, this gives an isomorphism test for graphs of Euler genus at most g running in time n^O((log g)^c)
A Spectral Assignment Approach for the Graph Isomorphism Problem
In this paper, we propose algorithms for the graph isomorphism (GI) problem
that are based on the eigendecompositions of the adjacency matrices. The
eigenvalues of isomorphic graphs are identical. However, two graphs and
can be isospectral but non-isomorphic. We first construct a graph
isomorphism testing algorithm for friendly graphs and then extend it to
unambiguous graphs. We show that isomorphisms can be detected by solving a
linear assignment problem. If the graphs possess repeated eigenvalues, which
typically correspond to graph symmetries, finding isomorphisms is much harder.
By repeatedly perturbing the adjacency matrices and by using properties of
eigenpolytopes, it is possible to break symmetries of the graphs and
iteratively assign vertices of to vertices of , provided that an
admissible assignment exists. This heuristic approach can be used to construct
a permutation which transforms into if the graphs are
isomorphic. The methods will be illustrated with several guiding examples
On the equivalence between graph isomorphism testing and function approximation with GNNs
Graph neural networks (GNNs) have achieved lots of success on
graph-structured data. In the light of this, there has been increasing interest
in studying their representation power. One line of work focuses on the
universal approximation of permutation-invariant functions by certain classes
of GNNs, and another demonstrates the limitation of GNNs via graph isomorphism
tests.
Our work connects these two perspectives and proves their equivalence. We
further develop a framework of the representation power of GNNs with the
language of sigma-algebra, which incorporates both viewpoints. Using this
framework, we compare the expressive power of different classes of GNNs as well
as other methods on graphs. In particular, we prove that order-2 Graph
G-invariant networks fail to distinguish non-isomorphic regular graphs with the
same degree. We then extend them to a new architecture, Ring-GNNs, which
succeeds on distinguishing these graphs and provides improvements on real-world
social network datasets
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