1,698 research outputs found
Isomorphism test for digraphs with weighted edges
Colour refinement is at the heart of all the most efficient graph isomorphism software packages. In this paper we present a method for extending the applicability of refinement algorithms to directed graphs with weighted edges. We use Traces as a reference software, but the proposed solution is easily transferrable to any other refinement-based graph isomorphism tool in the literature. We substantiate the claim that the performances of the original algorithm remain substantially unchanged by showing experiments for some classes of benchmark graphs
A Linear Upper Bound on the Weisfeiler-Leman Dimension of Graphs of Bounded Genus
The Weisfeiler-Leman (WL) dimension of a graph is a measure for the inherent descriptive complexity of the graph. While originally derived from a combinatorial graph isomorphism test called the Weisfeiler-Leman algorithm, the WL dimension can also be characterised in terms of the number of variables that is required to describe the graph up to isomorphism in first-order logic with counting quantifiers.
It is known that the WL dimension is upper-bounded for all graphs that exclude some fixed graph as a minor [M. Grohe, 2017]. However, the bounds that can be derived from this general result are astronomic. Only recently, it was proved that the WL dimension of planar graphs is at most 3 [S. Kiefer et al., 2017].
In this paper, we prove that the WL dimension of graphs embeddable in a surface of Euler genus g is at most 4g+3. For the WL dimension of graphs embeddable in an orientable surface of Euler genus g, our approach yields an upper bound of 2g + 3
The Iteration Number of Colour Refinement
The Colour Refinement procedure and its generalisation to higher dimensions, the Weisfeiler-Leman algorithm, are central subroutines in approaches to the graph isomorphism problem. In an iterative fashion, Colour Refinement computes a colouring of the vertices of its input graph.
A trivial upper bound on the iteration number of Colour Refinement on graphs of order n is n-1. We show that this bound is tight. More precisely, we prove via explicit constructions that there are infinitely many graphs G on which Colour Refinement takes |G|-1 iterations to stabilise. Modifying the infinite families that we present, we show that for every natural number n ? 10, there are graphs on n vertices on which Colour Refinement requires at least n-2 iterations to reach stabilisation
Comparative Design-Choice Analysis of Color Refinement Algorithms Beyond the Worst Case
Color refinement is a crucial subroutine in symmetry detection in theory as well as practice. It has further applications in machine learning and in computational problems from linear algebra.
While tight lower bounds for the worst case complexity are known [Berkholz, Bonsma, Grohe, ESA2013] no comparative analysis of design choices for color refinement algorithms is available.
We devise two models within which we can compare color refinement algorithms using formal methods, an online model and an approximation model. We use these to show that no online algorithm is competitive beyond a logarithmic factor and no algorithm can approximate the optimal color refinement splitting scheme beyond a logarithmic factor.
We also directly compare strategies used in practice showing that, on some graphs, queue based strategies outperform stack based ones by a logarithmic factor and vice versa. Similar results hold for strategies based on priority queues
Dimension Reduction via Colour Refinement
Colour refinement is a basic algorithmic routine for graph isomorphism
testing, appearing as a subroutine in almost all practical isomorphism solvers.
It partitions the vertices of a graph into "colour classes" in such a way that
all vertices in the same colour class have the same number of neighbours in
every colour class. Tinhofer (Disc. App. Math., 1991), Ramana, Scheinerman, and
Ullman (Disc. Math., 1994) and Godsil (Lin. Alg. and its App., 1997)
established a tight correspondence between colour refinement and fractional
isomorphisms of graphs, which are solutions to the LP relaxation of a natural
ILP formulation of graph isomorphism.
We introduce a version of colour refinement for matrices and extend existing
quasilinear algorithms for computing the colour classes. Then we generalise the
correspondence between colour refinement and fractional automorphisms and
develop a theory of fractional automorphisms and isomorphisms of matrices.
We apply our results to reduce the dimensions of systems of linear equations
and linear programs. Specifically, we show that any given LP L can efficiently
be transformed into a (potentially) smaller LP L' whose number of variables and
constraints is the number of colour classes of the colour refinement algorithm,
applied to a matrix associated with the LP. The transformation is such that we
can easily (by a linear mapping) map both feasible and optimal solutions back
and forth between the two LPs. We demonstrate empirically that colour
refinement can indeed greatly reduce the cost of solving linear programs
WL meet VC
Recently, many works studied the expressive power of graph neural networks
(GNNs) by linking it to the -dimensional Weisfeiler--Leman algorithm
(). Here, the is a well-studied
heuristic for the graph isomorphism problem, which iteratively colors or
partitions a graph's vertex set. While this connection has led to significant
advances in understanding and enhancing GNNs' expressive power, it does not
provide insights into their generalization performance, i.e., their ability to
make meaningful predictions beyond the training set. In this paper, we study
GNNs' generalization ability through the lens of Vapnik--Chervonenkis (VC)
dimension theory in two settings, focusing on graph-level predictions. First,
when no upper bound on the graphs' order is known, we show that the bitlength
of GNNs' weights tightly bounds their VC dimension. Further, we derive an upper
bound for GNNs' VC dimension using the number of colors produced by the
. Secondly, when an upper bound on the graphs' order is
known, we show a tight connection between the number of graphs distinguishable
by the and GNNs' VC dimension. Our empirical study
confirms the validity of our theoretical findings.Comment: arXiv admin note: text overlap with arXiv:2206.1116
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