Related papers: The probabilistic Weisfeiler-Leman algorithm
A pair of complementary algorithms are presented. One of the pair is a fast method for connecting graphs with an edge. The other is a fast method for removing edges from a graph. Both algorithms employ the same tree based graph…
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…
The $k$-dimensional Weisfeiler-Leman ($k$-WL) algorithm is a simple combinatorial algorithm that was originally designed as a graph isomorphism heuristic. It naturally finds applications in Babai's quasipolynomial time isomorphism…
We present a randomized algorithm that, given a constant $\epsilon > 0$, outputs a proper $(1+\epsilon)\Delta$-edge-coloring of an $m$-edge simple graph $G$ of maximum degree $\Delta \geq 1/\epsilon$ in $O(m)$ time with high probability.…
Let $\epsilon \in (0, 1)$ and $n, \Delta \in \mathbb N$ be such that $\Delta = \Omega\left(\max\left\{\frac{\log n}{\epsilon},\, \left(\frac{1}{\epsilon}\log \frac{1}{\epsilon}\right)^2\right\}\right)$. Given an $n$-vertex $m$-edge simple…
The Weisfeiler-Leman dimension of a graph $G$ is the least number $k$ such that the $k$-dimensional Weisfeiler-Leman algorithm distinguishes $G$ from every other non-isomorphic graph. The dimension is a standard measure of the descriptive…
The combinatorial refinement techniques have proven to be an efficient approach to isomorphism testing for particular classes of graphs. If the number of refinement rounds is small, this puts the corresponding isomorphism problem in a…
We show that on graphs with n vertices, the 2-dimensional Weisfeiler-Leman algorithm requires at most O(n^2/log(n)) iterations to reach stabilization. This in particular shows that the previously best, trivial upper bound of O(n^2) is…
The $k$-dimensional Weisfeiler-Leman algorithm is a powerful tool in graph isomorphism testing. For an input graph $G$, the algorithm determines a canonical coloring of $s$-tuples of vertices of $G$ for each $s$ between 1 and $k$. We say…
The Weisfeiler-Leman (WL) algorithm is a well-known combinatorial procedure for detecting symmetries in graphs and it is widely used in graph-isomorphism tests. It proceeds by iteratively refining a colouring of vertex tuples. The number of…
The Weisfeiler-Leman (WL) algorithm is a combinatorial procedure that computes colorings on graphs, which can often be used to detect their (non-)isomorphism. Particularly the 1- and 2-dimensional versions 1-WL and 2-WL have received much…
We present a simple $(1+\varepsilon)\Delta$-edge-coloring algorithm for graphs of maximum degree $\Delta = \Omega(\log n / \varepsilon)$ with running time $O\left(m\,\log^3 n/\varepsilon^3\right)$. Our algorithm improves upon that of [Duan,…
In recent years, algorithms and neural architectures based on the Weisfeiler-Leman algorithm, a well-known heuristic for the graph isomorphism problem, emerged as a powerful tool for (supervised) machine learning with graphs and relational…
The Weisfeiler-Leman procedure is a widely-used technique for graph isomorphism testing that works by iteratively computing an isomorphism-invariant coloring of vertex tuples. Meanwhile, a fundamental tool in structural graph theory, which…
We give a randomized algorithm that properly colors the vertices of a triangle-free graph G on n vertices using O(\Delta(G)/ log \Delta(G)) colors, where \Delta(G) is the maximum degree of G. The algorithm takes O(n\Delta2(G)log\Delta(G))…
We show that the 2-dimensional Weisfeiler-Leman algorithm stabilizes n-vertex graphs after at most O(n log n) iterations. This implies that if such graphs are distinguishable in 3-variable first order logic with counting, then they can also…
The vertex coloring problem to find chromatic numbers is known to be unsolvable in polynomial time. Although various algorithms have been proposed to efficiently compute chromatic numbers, they tend to take an enormous amount of time for…
A circulant graph is a Cayley graph of a finite cyclic group. The Weisfeiler-Leman-dimension of a circulant graph $X$ with respect to the class of all circulant graphs is the smallest positive integer~$m$ such that the $m$-dimensional…
The classical Weisfeiler-Leman algorithm aka color refinement is fundamental for graph learning with kernels and neural networks. Originally developed for graph isomorphism testing, the algorithm iteratively refines vertex colors. On many…
We present the first polynomial-time algorithm to exactly compute the number of labeled chordal graphs on $n$ vertices. Our algorithm solves a more general problem: given $n$ and $\omega$ as input, it computes the number of…