Reduction Techniques for Graph Isomorphism in the Context of Width Parameters

We study the parameterized complexity of the graph isomorphism problem when parameterized by width parameters related to tree decompositions. We apply the following technique to obtain fixed-parameter tractability for such parameters. We first compute an isomorphism invariant set of potential bags for a decomposition and then apply a restricted version of the Weisfeiler-Lehman algorithm to solve isomorphism. With this we show fixed-parameter tractability for several parameters and provide a unified explanation for various isomorphism results concerned with parameters related to tree decompositions. As a possibly first step towards intractability results for parameterized graph isomorphism we develop an fpt Turing-reduction from strong tree width to the a priori unrelated parameter maximum degree.Comment: 23 pages, 4 figure

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

Fixed-parameter tractable canonization and isomorphism test for graphs of bounded treewidth

We give a fixed-parameter tractable algorithm that, given a parameter $k$ and two graphs $G_1,G_2$, either concludes that one of these graphs has treewidth at least $k$, or determines whether $G_1$ and $G_2$ are isomorphic. The running time of the algorithm on an $n$-vertex graph is $2^{O(k^5\log k)}\cdot n^5$, and this is the first fixed-parameter algorithm for Graph Isomorphism parameterized by treewidth. Our algorithm in fact solves the more general canonization problem. We namely design a procedure working in $2^{O(k^5\log k)}\cdot n^5$ time that, for a given graph $G$ on $n$ vertices, either concludes that the treewidth of $G$ is at least $k$, or: * finds in an isomorphic-invariant way a graph $\mathfrak{c}(G)$ that is isomorphic to $G$; * finds an isomorphism-invariant construction term --- an algebraic expression that encodes $G$ together with a tree decomposition of $G$ of width $O(k^4)$. Hence, the isomorphism test reduces to verifying whether the computed isomorphic copies or the construction terms for $G_1$ and $G_2$ are equal.Comment: Full version of a paper presented at FOCS 201

Courcelle's Theorem - A Game-Theoretic Approach

Courcelle's Theorem states that every problem definable in Monadic Second-Order logic can be solved in linear time on structures of bounded treewidth, for example, by constructing a tree automaton that recognizes or rejects a tree decomposition of the structure. Existing, optimized software like the MONA tool can be used to build the corresponding tree automata, which for bounded treewidth are of constant size. Unfortunately, the constants involved can become extremely large - every quantifier alternation requires a power set construction for the automaton. Here, the required space can become a problem in practical applications. In this paper, we present a novel, direct approach based on model checking games, which avoids the expensive power set construction. Experiments with an implementation are promising, and we can solve problems on graphs where the automata-theoretic approach fails in practice.Comment: submitte