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

Compact graphs and equitable partitions

AbstractLet G be a graph with adjacency matrix A, and let Γ be the set of all permutation matrices which commute with A. We call G compact if every doubly stochastic matrix which commutes with A is a convex combination of matrices from Γ. We characterize the graphs for which S(A) = {I} and show that the automorphism group of a compact regular graph is generously transitive, i.e., given any two vertices, there is an automorphism which interchanges them. We also describe a polynomial time algorithm for determining whether a regular graph on a prime number of vertices is compact

The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs

In this paper we study the complexity of the following problems: 1. Given a colored graph X=(V,E,c), compute a minimum cardinality set of vertices S (subset of V) such that no nontrivial automorphism of X fixes all vertices in S. A closely related problem is computing a minimum base S for a permutation group G <= S_n given by generators, i.e., a minimum cardinality subset S of [n] such that no nontrivial permutation in G fixes all elements of S. Our focus is mainly on the parameterized complexity of these problems. We show that when k=|S| is treated as parameter, then both problems are MINI[1]-hard. For the dual problems, where k=n-|S| is the parameter, we give FPT~algorithms. 2. A notion closely related to fixing is called individualization. Individualization combined with the Weisfeiler-Leman procedure is a fundamental technique in algorithms for Graph Isomorphism. Motivated by the power of individualization, in the present paper we explore the complexity of individualization: what is the minimum number of vertices we need to individualize in a given graph such that color refinement "succeeds" on it. Here "succeeds" could have different interpretations, and we consider the following: It could mean the individualized graph becomes: (a) discrete, (b) amenable, (c)compact, or (d) refinable. In particular, we study the parameterized versions of these problems where the parameter is the number of vertices individualized. We show a dichotomy: For graphs with color classes of size at most 3 these problems can be solved in polynomial time, while starting from color class size 4 they become W[P]-hard

The Weisfeiler-Leman dimension of conjunctive queries

A graph parameter is a function on graphs with the property that, for any pair of isomorphic graphs 1 and 2, (1) = (2). The Weisfeiler–Leman (WL) dimension of is the minimum such that, if 1 and 2 are indistinguishable by the -dimensional WL-algorithm then (1) = (2). The WL-dimension of is ∞ if no such exists. We study the WL-dimension of graph parameters characterised by the number of answers from a fixed conjunctive query to the graph. Given a conjunctive query , we quantify the WL-dimension of the function that maps every graph to the number of answers of in . The works of Dvorák (J. Graph Theory 2010), Dell, Grohe, and Rattan (ICALP 2018), and Neuen (ArXiv 2023) have answered this question for full conjunctive queries, which are conjunctive queries without existentially quantified variables. For such queries , the WL-dimension is equal to the treewidth of the Gaifman graph of . In this work, we give a characterisation that applies to all conjunctive queries. Given any conjunctive query , we prove that its WL-dimension is equal to the semantic extension width sew(), a novel width measure that can be thought of as a combination of the treewidth of and its quantified star size, an invariant introduced by Durand and Mengel (ICDT 2013) describing how the existentially quantified variables of are connected with the free variables. Using the recently established equivalence between the WL-algorithm and higher-order Graph Neural Networks (GNNs) due to Morris et al. (AAAI 2019), we obtain as a consequence that the function counting answers to a conjunctive query cannot be computed by GNNs of order smaller than sew(). The majority of the paper is concerned with establishing a lower bound of the WL-dimension of a query. Given any conjunctive query with semantic extension width , we consider a graph of treewidth obtained from the Gaifman graph of by repeatedly cloning the vertices corresponding to existentially quantified variables. Using a modification due to Fürer (ICALP 2001) of the Cai-Fürer-Immerman construction (Combinatorica 1992), we then obtain a pair of graphs ( ) and ˆ( ) that are indistinguishable by the ( − 1)- dimensional WL-algorithm since has treewidth . Finally, in the technical heart of the paper, we show that has a different number of answers in ( ) and ˆ( ). Thus, can distinguish two graphs that cannot be distinguished by the ( − 1)-dimensional WL-algorithm, so the WL-dimension of is at least