Colour Refinement: A Simple Partitioning Algorithm with Applications From Graph Isomorphism Testing to Machine Learning (Invited Talk)
نویسنده
چکیده
Colour refinement is a simple algorithm that partitions the vertices of a graph according their “iterated degree sequence.” It has very efficient implementations, running in quasilinear time, and a surprisingly wide range of applications. The algorithm has been designed in the context of graph isomorphism testing, and it is used an important subroutine in almost all practical graph isomorphism tools. Somewhat surprisingly, other applications in machine learning, probabilistic inference, and linear programming have surfaced recently. In the first part of my talk, I will introduce the basic algorithm as well as higher dimensional extensions known as the k-dimensional Weisfeiler-Lehman algorithm. I will also discuss an unexpected connection between colour refinement and a natural linear programming approach to graph isomorphism testing. In the second part of my talk, I will discuss various applications of colour refinement. 1998 ACM Subject Classification F.2 Analysis of Algorithms and Problem Complexity
منابع مشابه
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 [27], Ramana, Scheinerman, and Ullman [23] and Godsil [12] est...
متن کاملGraph isomorphism testing boosted by path coloring
A method for improving the efficiency of graph isomorphism testing is presented. The method uses the structure of the graph colored by vertex hash codes as a means of partitioning vertices into equivalence classes, which in turn reduces the combinatorial burden of isomorphism testing. Unrolling the graph into a tree at each vertex allows structurally different regular graphs to be discriminated...
متن کاملAn exponential lower bound for Individualization-Refinement algorithms for Graph Isomorphism
The individualization-refinement paradigm provides a strong toolbox for testing isomorphism of two graphs and indeed, the currently fastest implementations of isomorphism solvers all follow this approach. While these solvers are fast in practice, from a theoretical point of view, no general lower bounds concerning the worst case complexity of these tools are known. In fact, it is an open questi...
متن کاملIsomorphism, Automorphism Partitioning, and Canonical Labeling Can Be Solved in Polynomial-Time for Molecular Graphs
The graph isomorphism problem belongs to the class of NP problems, and has been conjectured intractable, although probably not NP-complete. However, in the context of chemistry, because molecules are a restricted class of graphs, the problem of graph isomorphism can be solved efficiently (i.e., in polynomial-time). This paper presents the theoretical results that for all molecules, the problems...
متن کاملMaximin affinity learning of image segmentation
Images can be segmented by first using a classifier to predict an affinity graph that reflects the degree to which image pixels must be grouped together and then partitioning the graph to yield a segmentation. Machine learning has been applied to the affinity classifier to produce affinity graphs that are good in the sense of minimizing edge misclassification rates. However, this error measure ...
متن کامل