Isomorphism Problem Research Articles

Isomorphism Problems and Groups of Automorphisms for Ore Extensions $$K[x][y; f\frac{d}{dx} ]$$ (Prime Characteristic)

AbstractLet $$\Lambda (f) = K[x][y; f\frac{d}{dx} ]$$ Λ ( f ) = K [ x ] [ y ; f d dx ] be an Ore extension of a polynomial algebra K[x] over an arbitrary field K of characteristic $$p>0$$ p > 0 where $$f\in K[x]$$ f ∈ K [ x ] . For each polynomial f, the automorphism group of the algebras $$\Lambda (f)$$ Λ ( f ) is explicitly described. The automorphism group $$\textrm{Aut}_K(\Lambda (f))=\mathbb {S}\rtimes G_f$$ Aut K ( Λ ( f ) ) = S ⋊ G f is a semidirect product of two explicit groups where $$G_f$$ G f is the eigengroup of the polynomial f (the set of all automorphisms of K[x] such that f is their common eigenvector). For each polynomial f, the eigengroup $$G_f$$ G f is explicitly described. It is proven that every subgroup of $$\textrm{Aut}_K(K[x])$$ Aut K ( K [ x ] ) is the eigengroup of a polynomial. It is proven that the Krull and global dimensions of the algebra $$\Lambda (f)$$ Λ ( f ) are 2. The prime, completely prime, primitive and maximal ideals of the algebra $$\Lambda (f)$$ Λ ( f ) are classified.

Decidability of the isomorphism problem between multidimensional substitutive subshifts

Abstract An important question in dynamical systems is the classification problem, that is, the ability to distinguish between two isomorphic systems. In this work, we study the topological factors between a family of multidimensional substitutive subshifts generated by morphisms with uniform support. We prove that it is decidable to check whether two minimal aperiodic substitutive subshifts are isomorphic. The strategy followed in this work consists of giving a complete description of the factor maps between these subshifts. Then, we deduce some interesting consequences on coalescence, automorphism groups, and the number of aperiodic symbolic factors of substitutive subshifts. We also prove other combinatorial results on these substitutions, such as the decidability of defining a subshift, the computability of the constant of recognizability, and the conjugacy between substitutions with different supports.

Efficient computations in central simple algebras using Amitsur cohomology

We introduce a presentation for central simple algebras over a field k using Amitsur cohomology. We provide efficient algorithms for computing a cocycle corresponding to any such algebra given by structure constants. If k is a number field, we use this presentation to prove that the explicit isomorphism problem (i.e., finding an isomorphism between central simple algebras given by structure constants) reduces to S-unit group computation and other related number theoretical computational problems. This also yields, conditionally on the generalised Riemann hypothesis, the first polynomial quantum algorithm for the explicit isomorphism problem over number fields.

The Modular Isomorphism Problem – the alternative perspective on counterexamples

As a result of impressive research [5], D. García-Lucas, Á. del Río and L. Margolis defined an infinite series of non-isomorphic 2-groups G and H, whose group algebras FG and FH over the field F=F2 are isomorphic, solving negatively the long-standing Modular Isomorphism Problem (MIP). In this note we give a different perspective on their examples and show that they are special cases of a more general construction. We also show that this type of construction for p>2 does not provide a similar counterexample to the MIP.

On the twisted group ring isomorphism problem for a class of groups

The twisted group ring isomorphism problem (TGRIP) is a variation of the classical group ring isomorphism problem. It asks whether the ring structure of the twisted group ring determines the group up to isomorphism. In this article, we study the TGRIP for direct product and central product of groups. We provide some criteria to answer the TGRIP for groups by answering the TGRIP for certain associated quotients. As an application of these results, we provide several examples. Finally, we answer the TGRIP for extra-special p-groups, and for all but five groups of order p5, where p≥5 is prime.

State of the Art and Potentialities of Graph-level Learning

Graphs have a superior ability to represent relational data, such as chemical compounds, proteins, and social networks. Hence, graph-level learning, which takes a set of graphs as input, has been applied to many tasks, including comparison, regression, classification, and more. Traditional approaches to learning a set of graphs heavily rely on hand-crafted features, such as substructures. While these methods benefit from good interpretability, they often suffer from computational bottlenecks, as they cannot skirt the graph isomorphism problem. Conversely, deep learning has helped graph-level learning adapt to the growing scale of graphs by extracting features automatically and encoding graphs into low-dimensional representations. As a result, these deep graph learning methods have been responsible for many successes. Yet, no comprehensive survey reviews graph-level learning starting with traditional learning and moving through to the deep learning approaches. This article fills this gap and frames the representative algorithms into a systematic taxonomy covering traditional learning, graph-level deep neural networks, graph-level graph neural networks, and graph pooling. In addition, the evolution and interaction between methods from these four branches within their developments are examined to provide an in-depth analysis. This is followed by a brief review of the benchmark datasets, evaluation metrics, and common downstream applications. Finally, the survey concludes with an in-depth discussion of 12 current and future directions in this booming field.

Portable Network Resolving Huge-Graph Isomorphism Problem

Abstract The graph isomorphism, as a key task in graph data analysis, is of great significance for the understanding, feature extraction, and pattern recognition of graph data. The best performance of traditional methods is the quasi-polynomial time complexity, which is infeasible for huge graphs. This paper aims to propose a lightweight graph network to resolve the problem of isomorphism in huge graphs. We propose a partitioning algorithm with linear time complexity based on isomorphic necessary condition to handle network graphs that cannot be fully computed by a single computer. We use anti-weight convolution analysis and skip connections to obtain a more stable representation with fewer layers and parameters. We implement simulations on personal computer (PC) with graphs consisting of hundred millions of edges, demonstrating the identification performance ($>98\%$) and time efficiency.

PathLAD+: Towards effective exact methods for subgraph isomorphism problem

The subgraph isomorphism problem (SIP) is a challenging problem with wide practical applications. In the last decade, despite being a theoretical hard problem, researchers designed various algorithms for solving SIP. In this work, we propose five main strategies and develop an improved exact algorithm for SIP. First, we design a probing search procedure to try whether the search procedure can successfully obtain a solution at first sight. Second, we design a novel matching ordering strategy as a value-ordering heuristic, which uses some useful information obtained from the probing search procedure to preferentially select some promising target vertices. Third, we discuss the characteristics of different propagation methods in the context of SIP and present an adaptive propagation method to make a good balance between these methods. Moreover, to further improve the performance of solving large graphs, we propose an enhanced implementation of the edge constraint method and a domain limitation strategy, which aims to accelerate the search process. Experimental results on a broad range of classic and graph-database benchmarks show that our proposed algorithm performs better than several state-of-the-art algorithms for the SIP.

Automorphisms of Graph Products of Groups and Acylindrical Hyperbolicity

This article is dedicated to the study of the acylindrical hyperbolicity of automorphism groups of graph products of groups. Our main result is that, if Γ \Gamma is a finite graph which contains at least two vertices and is not a join and if G \mathcal {G} is a collection of finitely generated irreducible groups, then either Γ G \Gamma \mathcal {G} is infinite dihedral or A u t ( Γ G ) \mathrm {Aut}(\Gamma \mathcal {G}) is acylindrically hyperbolic. This theorem is new even for right-angled Artin groups and right-angled Coxeter groups. Various consequences are deduced from this statement and from the techniques used to prove it. For instance, we show that the automorphism groups of most graph products verify vastness properties such as being SQ-universal; we show that many automorphism groups of graph products do not satisfy Kazhdan’s property (T); we solve the isomorphism problem between graph products in some cases; and we show that a graph product of coarse median groups, as defined by Bowditch, is coarse median itself.

The isomorphism problem for small-rose generalized Baumslag-Solitar groups

A generalized Baumslag-Solitar group is a finitely generated group that acts on a tree with infinite-cyclic vertex and edge stabilizers. In this paper, we show that the isomorphism problem is solvable for small-rose non-ascending generalized Baumslag-Solitar groups.

The Complexity of Subelection Isomorphism Problems

We study extensions of the Election Isomorphism problem, focused on the existence of isomorphic subelections. Specifically, we propose the Subelection Isomorphism and the Maximum Common Subelection problems and study their computational complexity and approximability. Using our problems in experiments, we provide some insights into the nature of several statistical models of elections

Automorphisms and isomorphisms of maps in linear time

A map is a \(2\) -cell decomposition of a closed compact surface, i.e., an embedding of a graph such that every face is homeomorphic to an open disc. An automorphism of a map can be thought of as a permutation of the vertices which preserves the vertex-edge-face incidences in the embedding. Every automorphism of a map determines an angle-preserving homeomorphism of the surface. While it is conjectured that there is no “truly subquadratic” algorithm for testing map isomorphism for unconstrained genus, we present a linear-time algorithm for computing the generators of the automorphism group of a map on an orientable surface of genus \(g\neq 0\) , parametrized by the genus \(g\) . A map on an orientable surface is uniform if the cyclic vector of sizes of faces incident to a vertex \(v\) does not depend on the choice of \(v\) . The algorithm applies a sequence of local reductions and produces a uniform map, while preserving the automorphism group. The automorphism group of the original map can be reconstructed from the automorphism group of the associated uniform map in linear time. We also extend the algorithm to non-orientable surfaces by making use of the antipodal double-cover. The algorithm can be used to solve the map isomorphism problem between maps (orientable or non-orientable) of bounded negative Euler characteristic.

On the modular isomorphism problem for 2-generated groups with cyclic derived subgroup

We continue the analysis of the modular isomorphism problem for [Formula: see text]-generated [Formula: see text]-groups with cyclic derived subgroup, [Formula: see text], started in [D. García-Lucas, Á. del Río and M. Stanojkowski, On group invariants determined by modular group algebras: Even versus odd characteristic, Algebra Represent. Theory 26 (2022) 2683–2707, doi:10.1007/s10468-022-10182-x]. We show that if [Formula: see text] belongs to this class of groups, then the isomorphism type of the quotients [Formula: see text] and [Formula: see text] are determined by its modular group algebra. In fact, we obtain a more general but technical result, expressed in terms of the classification [O. Broche, D. García-Lucas and Á. del Río, A classification of the finite 2-generator cyclic-by-abelian groups of prime-power order, Int. J. Algebra Comput. 33(4) (2023) 641–686]. We also show that for groups in this class of order at most [Formula: see text], the modular isomorphism problem has positive answer. Finally, we describe some families of groups of order [Formula: see text] whose group algebras over the field with [Formula: see text] elements cannot be distinguished with the techniques available to us.

The codegree isomorphism problem for finite simple groups

The codegree isomorphism problem for finite simple groups

A Framework to Improve the Implementations of Linear Layers

This paper presents a novel approach to optimizing the linear layer of block ciphers using the matrix decomposition framework. It is observed that the reduction properties proposed by Xiang et al. (in FSE 2020) need to be improved. To address these limitations, we propose a new reduction framework with a complete reduction algorithm and swapping algorithm. Our approach formulates matrix decomposition as a new framework with an adaptive objective function and converts the problem to a Graph Isomorphism problem (GI problem). Using the new reduction algorithm, we were able to achieve lower XOR counts and depths of quantum implementations under the s-XOR metric. Our results outperform previous works for many linear layers of block ciphers and hash functions; some of them are better than the current g-XOR implementation. For the AES MixColumn operation, we get two implementations with 91 XOR counts and depth 13 of in-place quantum implementation, respectively.

Elimination Distance to Bounded Degree on Planar Graphs Preprint

We study the graph parameter elimination distance to bounded degree, which was introduced by Bulian and Dawar in their study of the parameterized complexity of the graph isomorphism problem. We prove that the problem is fixed-parameter tractable on planar graphs, that is, there exists an algorithm that given a planar graph G and integers d, k decides in time f(k, d) · nc for a computable function f and constant c whether the elimination distance of G to the class of degree d graphs is at most k.

On isomorphisms to a free group and beyond

The isomorphism problem for infinite finitely presented groups is probably the hardest among standard algorithmic problems in group theory. Classes of groups where it has been completely solved are nilpotent groups, hyperbolic groups, and limit groups. In this short paper, we address the problem of isomorphism to particular groups, including free groups. We also address the algorithmic problem of embedding a finitely presented group in a given limit group.

The isomorphism problem for oligomorphic groups with weak elimination of imaginaries

AbstractIn Kechris et al. [J. Symb. Log. 83 (2018), no. 3, 1190–1203], it was asked if equality on the reals is sharp as a lower bound for the complexity of topological isomorphism between oligomorphic groups. We prove that under the assumption of weak elimination of imaginaries, this is indeed the case. Our methods are model theoretic and they also have applications on the classical problem of reconstruction of isomorphisms of permutation groups from (topological) isomorphisms of automorphisms groups. As a concrete application, we give an explicit description of for any vector space of dimension over a finite field, in affinity with the classical description for finite‐dimensional spaces due to Schreier and van der Waerden.

Relationships between almost completely decomposable abelian groups and their multiplication groups

Let [Formula: see text] be the class of Abelian block-rigid almost completely decomposable groups of ring type with cyclic regulator quotient. In this paper, we study relationships between the above groups [Formula: see text] and their multiplication groups Mult [Formula: see text]. It is proved that groups from [Formula: see text] are determined by their multiplication groups. For a rigid group [Formula: see text], the isomorphism problem is solved: we describe multiplications from [Formula: see text] that define isomorphic rings on [Formula: see text]. We also describe Abelian groups that are realized as the multiplication group of some group in [Formula: see text]. We describe groups in [Formula: see text] that are isomorphic to their multiplication groups.

BF-BigGraph: An efficient subgraph isomorphism approach using machine learning for big graph databases

Graph databases are flexible NoSQL databases used to efficiently store and query complex and big data. One of the most difficult problems in graph databases is the problem of subgraph isomorphism, which involves finding a matching pattern in a given graph. Subgraph isomorphism algorithms generally encounter problems in the efficient processing of complex queries based on a lack of pruning methods and the use of a matching order. In this study, we present a new subgraph isomorphism approach based on the best-first search design strategy and name it BF-BigGraph. Our approach includes a machine learning technique to efficiently find the best matching order for various complex queries. The parameters we used in our approach as heuristics to improve the performance of complex queries on graph-based NoSQL databases are database volatility, database size, type of query, and the size of the query. We utilized the Random Forest machine learning method to narrow candidate nodes to a higher level of search and effectively reduce the search space for efficient querying and retrieval. We compared BF-BigGraph with state-of-the-art approaches, namely BB-Graph, Neo4j’s Cypher, DualIso, GraphQL, TurboIso, and VF3 using publicly available databases including undirected graphs; WorldCup, Pokec, Youtube, and a big graph database of a real demographic application (a population database) with approximately 70 million nodes of a big directed graph. The performance results of our approach for different types of complex queries on all these databases are significantly better in terms of computation time and required memory than other competing approaches in the literature.