Constructing Hard Examples for Graph Isomorphism.

We describe a method for generating graphs that provide difficult examples for practical Graph Isomorphism testers. We first give the theoretical construction, showing that we can have a family of graphs without any non-trivial automorphisms which also have high Weisfeiler-Leman dimension. The construction is based on properties of random 3XOR-formulas. We describe how to convert such a formula into a graph which has the desired properties with high probability. We validate the method by experimental implementations. We construct random formulas and validate them with a SAT solver to filter through suitable ones, and then convert them into graphs. Experimental results demonstrate that the resulting graphs do provide hard examples that match the hardest known benchmarks for graph isomorphism

Witnessed Symmetric Choice and Interpretations in Fixed-Point Logic with Counting

At the core of the quest for a logic for Ptime is a mismatch between algorithms making arbitrary choices and isomorphism-invariant logics. One approach to tackle this problem is witnessed symmetric choice. It allows for choices from definable orbits certified by definable witnessing automorphisms. We consider the extension of fixed-point logic with counting (IFPC) with witnessed symmetric choice (IFPC+WSC) and a further extension with an interpretation operator (IFPC+WSC+I). The latter operator evaluates a subformula in the structure defined by an interpretation. When similarly extending pure fixed-point logic (IFP), IFP+WSC+I simulates counting which IFP+WSC fails to do. For IFPC+WSC, it is unknown whether the interpretation operator increases expressiveness and thus allows studying the relation between WSC and interpretations beyond counting. In this paper, we separate IFPC+WSC from IFPC+WSC+I by showing that IFPC+WSC is not closed under FO-interpretations. By the same argument, we answer an open question of Dawar and Richerby regarding non-witnessed symmetric choice in IFP. Additionally, we prove that nesting WSC-operators increases the expressiveness using the so-called CFI graphs. We show that if IFPC+WSC+I canonizes a particular class of base graphs, then it also canonizes the corresponding CFI graphs. This differs from various other logics, where CFI graphs provide difficult instances

Witnessed Symmetric Choice and Interpretations in Fixed-Point Logic with Counting

At the core of the quest for a logic for PTime is a mismatch between algorithms making arbitrary choices and isomorphism-invariant logics. One approach to overcome this problem is witnessed symmetric choice. It allows for choices from definable orbits which are certified by definable witnessing automorphisms. We consider the extension of fixed-point logic with counting (IFPC) with witnessed symmetric choice (IFPC+WSC) and a further extension with an interpretation operator (IFPC+WSC+I). The latter operator evaluates a subformula in the structure defined by an interpretation. This structure possibly has other automorphisms exploitable by the WSC-operator. For similar extensions of pure fixed-point logic (IFP) it is known that IFP+WSCI simulates counting which IFP+WSC fails to do. For IFPC+WSC it is unknown whether the interpretation operator increases expressiveness and thus allows studying the relation between WSC and interpretations beyond counting. We separate IFPC+WSC from IFPC+WSCI by showing that IFPC+WSC is not closed under FO-interpretations. By the same argument, we answer an open question of Dawar and Richerby regarding non-witnessed symmetric choice in IFP. Additionally, we prove that nesting WSC-operators increases the expressiveness using the so-called CFI graphs. We show that if IFPC+WSC+I canonizes a particular class of base graphs, then it also canonizes the corresponding CFI graphs. This differs from various other logics, where CFI graphs provide difficult instances.Comment: 46 pages, 5 figures, [v2] and [v3] Corrected minor mistakes and added figure

Tree comparison: enumeration and application to cheminformatics

Graphs are a well-known data structure used in many application domains that rely on relationships between individual entities. Examples are social networks, where the users may be in friendship with each other, road networks, where one-way or bidirectional roads connect crossings, and work package assignments, where workers are assigned to tasks. In chem- and bioinformatics, molecules are often represented as molecular graphs, where vertices represent atoms, and bonds between them are represented by edges connecting the vertices. Since there is an ever-increasing amount of data that can be treated as graphs, fast algorithms are needed to compare such graphs. A well-researched concept to compare two graphs is the maximum common subgraph. On the one hand, this allows finding substructures that are common to both input graphs. On the other hand, we can derive a similarity score from the maximum common subgraph. A practical application is rational drug design which involves molecular similarity searches. In this thesis, we study the maximum common subgraph problem, which entails finding a largest graph, which is isomorphic to subgraphs of two input graphs. We focus on restrictions that allow polynomial-time algorithms with a low exponent. An example is the maximum common subtree of two input trees. We succeed in improving the previously best-known time bound. Additionally, we provide a lower time bound under certain assumptions. We study a generalization of the maximum common subtree problem, the block-and-bridge preserving maximum common induced subgraph problem between outerplanar graphs. This problem is motivated by the application to cheminformatics. First, the vast majority of drugs modeled as molecular graphs is outerplanar, and second, the blocks correspond to the ring structures and the bridges to atom chains or linkers. If we allow disconnected common subgraphs, the problem becomes NP-hard even for trees as input. We propose a second generalization of the maximum common subtree problem, which allows skipping vertices in the input trees while maintaining polynomial running time. Since a maximum common subgraph is not unique in general, we investigate the problem to enumerate all maximum solutions. We do this for both the maximum common subtree problem and the block-and-bridge preserving maximum common induced subgraph problem between outerplanar graphs. An arising subproblem which we analyze is the enumeration of maximum weight matchings in bipartite graphs. We support a weight function between the vertices and edges for all proposed common subgraph methods in this thesis. Thus the objective is to compute a common subgraph of maximum weight. The weights may be integral or real-valued, including negative values. A special case of using such a weight function is computing common subgraph isomorphisms between labeled graphs, where labels between mapped vertices and edges must be equal. An experimental study evaluates the practical running times and the usefulness of our block-and-bridge preserving maximum common induced subgraph algorithm against state of the art algorithms

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum