Isomorphism Problem Research Articles

Entanglement entropy in the ground state of supersymmetric fermion lattice models

In this paper, we study the entanglement entropy for strongly correlated spinless fermions in the ground state of a supersymmetric Hamiltonian on diverse graphs. Then, we use the entanglement entropy to study the graph isomorphism (GI) problem on some non-isomorphic pairs of graphs to distinguish them from each other. The GI problem is considered on pairs of non-isomorphic strongly regular graphs (SRG) and some cases of regular graphs. The SRGs are well-known for being highly symmetric, so that they rarely can be detected by classical and quantum algorithms. However, we show that most of the pairs of non-isomorphic SRGs can be distinguished by utilizing the supersymmetric entanglement entropy. Also, detecting the regular non-isomorphic graphs is often difficult. The obtained result illustrates that the entanglement entropy of the single-particle Hamiltonian cannot detect non-isomorphic pairs of regular graphs, but the entanglement entropy of the two-particle Hamiltonian can.

Recursive-Parallel Algorithm for Solving the Graph-Subgraph Isomorphism Problem

The paper proposes a parallel algorithm for solving the Graph-Subgraph Isomorphism Problem and makes an experimental study of its efficiency. The problem is one of the most famous NP-complete problems. Its solution may be required when solving many practical problems associated with the study of complex structures. We solve the problem in a formulation that requires finding all existing isomorphic substitutions or proving their absence. In view of the high complexity of the problem, it is natural to want to speed up its solution by parallelizing the algorithm.We used the RPM ParLib library, developed by the author, as the main tool to program the algorithm. This library allows us to develop effective applications for parallel computing on a local network under the control of the runtime environment .NET Framework. Thanks to this library, applications have the ability to generate parallel branches of computation directly during program execution and dynamically redistribute work between computing modules. Any language with support of the .NET Framework can be used as a programming language in conjunction with this library. For the numerical experiment, an open database from the Internet was used, which was specially developed to study algorithms for searching for isomorphic substitutions. Also, the author has developed a special application in C# for generating additional sets of initial data with specified characteristics. The aim of the experiment was to study the speedup achieved due to the recursively parallel organization of computations. The paper provides a detailed description of the proposed algorithm, as well as the results obtained during the experiment.

The isomorphism problem for universal enveloping algebras of solvable Lie algebras

This paper is a contribution to the isomorphism problem for universal enveloping algebras of finite-dimensional Lie algebras. We focus on solvable Lie algebras of small dimensions over fields of arbitrary characteristic. We prove, over an arbitrary field, that the isomorphism type of a metabelian Lie algebra whose derived subalgebra has codimension one is determined by its universal enveloping algebra. As an application of the results in this paper, we solve the isomorphism problem for solvable Lie algebras of dimension four over fields of characteristic zero and also point out the problems that occur in prime characteristic.

A correctly scaling rigorously spin-adapted and spin-complete open-shell CCSD implementation for arbitrary high-spin states.

In this paper, we report on a correctly scaling novel coupled cluster singles and doubles (CCSD) implementation for arbitrary high-spin open-shell states. The chosen cluster operator is completely spin-free, i.e., employs spatial substitutions only. It is composed of our recently developed Löwdin-type operators [N. Herrmann and M. Hanrath, J. Chem. Phys. 153, 164114 (2020)], which ensure (1) spin completeness and (2) spin adaption, i.e., spin purity of the CC wave function. In contrast to the proof-of-concept matrix-representation-based implementation presented there, the present implementation relies on second quantization and factorized tensor contractions. The generated singles and doubles operators are embedded in an equation generation engine. In the latter, Wick's theorem is used to derive prefactors arising from spin integration directly from the spin-free full contraction patterns. The obtained Wick terms composed of products of Kronecker deltas are represented by special non-antisymmetrized Goldstone diagrams. Identical (redundant) diagrams are identified by solving the underlying graph isomorphism problem. All non-redundant graphs are then automatically translated to locally-one term at a time-factorized tensor contractions. Finally, the spin-adapted and spin-complete (SASC) CCS and CCSD variants are applied to a set of small molecular test systems. Both correlation energies and amplitude norms hint toward a reasonable convergence of the SASC-CCSD method for a Baker-Campbell-Hausdorff series truncation of order four. In comparison to spin orbital CCSD, SASC-CCSD leads to slightly improved correlation energies with differences of up to 1.292mEH (1.10% with respect to full configuration identification) for quintet CH2 in the cc-pVDZ basis set.

The Power of the Weisfeiler--Leman Algorithm to Decompose Graphs

The Weisfeiler-Leman procedure is a widely-used technique for graph isomorphism testing that works by iteratively computing an isomorphism-invariant coloring of vertex tuples. Meanwhile, a fundamental tool in structural graph theory, which is often exploited in approaches to tackle the graph isomorphism problem, is the decomposition into 2- and 3-connected components. We prove that the 2-dimensional Weisfeiler-Leman algorithm implicitly computes the decomposition of a graph into its 3-connected components. This implies that the dimension of the algorithm needed to distinguish two given non-isomorphic graphs is at most the dimension required to distinguish non-isomorphic 3-connected components of the graphs (assuming dimension at least 2). To obtain our decomposition result, we show that, for k >= 2, the k-dimensional algorithm distinguishes k-separators, i.e., k-tuples of vertices that separate the graph, from other vertex k-tuples. As a byproduct, we also obtain insights about the connectivity of constituent graphs of association schemes. In an application of the results, we show the new upper bound of k on the Weisfeiler-Leman dimension of the class of graphs of treewidth at most k. Using a construction by Cai, F\"urer, and Immerman, we also provide a new lower bound that is asymptotically tight up to a factor of 2.

Envy-free allocations respecting social networks

Finding an envy-free allocation of indivisible resources to agents is a central task in many multiagent systems. Often, non-trivial envy-free allocations do not exist and determining whether they exist is computationally hard even under highly restricted settings. Classical envy-freeness requires that every agent likes the resources allocated to it at least as much as the resources allocated to any other agent. In many situations this assumption can be relaxed since the agents often do not even know each other. We enrich the envy-freeness concept by taking into account (directed) social networks of the agents. Thus, we require that every agent likes its own allocation at least as much as those of all its (out)neighbors. This leads to a “more local” concept of envy-freeness. We also consider a “strong” variant where every agent must like its own allocation more than those of all its (out)neighbors.We analyze the classical and the parameterized complexity of finding allocations that are complete and, at the same time, envy-free with respect to one of the variants of our new concept. To this end, we study different restrictions of the agents' preferences and of the social network structure. We identify cases that become easier (from Σ2P-hard or NP-hard to polynomial-time solvable) and cases that become harder (from polynomial-time solvable to NP-hard) when comparing classical envy-freeness with our graph envy-freeness. Furthermore, we spot cases where graph envy-freeness is easier to decide than strong graph envy-freeness, and vice versa. On the route to one of our fixed-parameter tractability results, we also establish a connection to a directed and colored variant of the classical Subgraph Isomorphism problem, thereby extending a known fixed-parameter tractability result for the latter.

On the isomorphism problem for some classes of computable algebraic structures

On the isomorphism problem for some classes of computable algebraic structures

Генерический полиномиальный алгоритм для проблемы изоморфизма двудольных графов

The isomorphism problem for bipartite graphs is as computationally hard as the well-known isomorphism problem for arbitrary finite graphs, for which no polynomial algorithms are known. The paper proposes a polynomial generic algorithm for the bipartite graph isomorphism problem, which gives the correct answer on the set of almost all inputs, and gives an undefined answer on the rare remaining inputs.

The isomorphism problem for tensor algebras of multivariable dynamical systems

AbstractWe resolve the isomorphism problem for tensor algebras of unital multivariable dynamical systems. Specifically, we show that unitary equivalence after a conjugation for multivariable dynamical systems is a complete invariant for complete isometric isomorphisms between their tensor algebras. In particular, this settles a conjecture of Davidson and Kakariadis,Inter. Math. Res. Not.2014(2014), 1289–1311 relating to work of Arveson,Acta Math.118(1967), 95–109 from the 1960s, and extends related work of Kakariadis and Katsoulis,J. Noncommut. Geom.8(2014), 771–787.

Isomorphisms of algebras of convolution operators

For $p,q\in [1,\infty)$, we study the isomorphism problem for the $p$- and $q$-convolution algebras associated to locally compact groups. While it is well known that not every group can be recovered from its group von Neumann algebra, we show that this is the case for the algebras $\mathrm{CV}_p(G)$ of $p$-convolvers and $\mathrm{PM}_p(G)$ of $p$-pseudomeasures, for $p\neq 2$. More generally, we show that if $\mathrm{CV}_p(G)$ is isometrically isomorphic to $\mathrm{CV}_q(H)$, with $p,q\neq 2$, then $G$ must be isomorphic to $H$ and $p$ and $q$ are either equal or conjugate. This implies that there is no $L^p$-version of Connes' uniqueness of the hyperfinite II$_1$-factor. Similar results apply to the algebra $\mathrm{PF}_p(G)$ of $p$-pseudofunctions, generalizing a classical result of Wendel. We also show that other $L^p$-rigidity results for groups can be easily recovered and extended using our main theorem. Our results answer questions originally formulated in the work of Herz in the 70's. Moreover, our methods reveal new information about the Banach algebras in question. As a non-trivial application, we verify the reflexivity conjecture for all Banach algebras lying between $\mathrm{PF}_p(G)$ and $\mathrm{CV}_p(G)$: if any such algebra is reflexive and amenable, then $G$ is finite.

DG algebra structures on the quantum affine n-space [formula omitted

Let A be a connected cochain DG algebra, whose underlying graded algebra A# is the quantum affine n-space O−1(kn). We compute all possible differential structures of A and show that there exists a one-to-one correspondence between{cochain DG algebraA|A#=O−1(kn)} and the n×n matrices Mn(k). For any M∈Mn(k), we write AO−1(k3)(M) for the DG algebra corresponding to it. We also study the isomorphism problems of these non-commutative DG algebras. For the cases n≤3, we check their homological properties. Unlike the case of n=2, we discover that not all of them are Calabi-Yau when n=3. In spite of this, we recognize those Calabi-Yau ones case by case. In brief, we solve the problem on how to judge whether a given such DG algebra AO−1(k3)(M) is Calabi-Yau.

A polynomial-time reduction from the multi-graph isomorphism problem to additive code equivalence

We present a polynomial time reduction from the multi-graph isomorphism problem to the problem of code equivalence of additive codes over finite extensions of ${\mathbb F}_2$.

Non-isomorphic 2-groups with isomorphic modular group algebras

Abstract We provide non-isomorphic finite 2-groups which have isomorphic group algebras over any field of characteristic 2, thus settling the Modular Isomorphism Problem.

Parallel Algorithm for Solving the Graph Isomorphism Problem

Parallel Algorithm for Solving the Graph Isomorphism Problem

Asymmetric coloring of locally finite graphs and profinite permutation groups: Tucker's Conjecture confirmed

An asymmetric coloring of a graph is a coloring of its vertices that is not preserved by any non-identity automorphism of the graph. The motion of a graph is the minimal degree of its automorphism group, i.e., the minimum number of elements that are moved (not fixed) by any non-identity automorphism. We confirm Tom Tucker's “Infinite Motion Conjecture” that connected locally finite graphs with infinite motion admit an asymmetric 2-coloring. We infer this from the more general result that the inverse limit of an infinite sequence of finite permutation groups with disjoint domains, viewed as a permutation group on the union of those domains, admits an asymmetric 2-coloring. The proof is based on the study of the interaction between epimorphisms of finite permutation groups and the structure of the setwise stabilizers of subsets of their domains. We note connections of the subject to computational group theory, asymptotic group theory, highly regular structures, and the Graph Isomorphism problem, and list a number of open problems.

An efficient procedure for mining egocentric temporal motifs

Temporal graphs are structures which model relational data between entities that change over time. Due to the complex structure of data, mining statistically significant temporal subgraphs, also known as temporal motifs, is a challenging task. In this work, we present an efficient technique for extracting temporal motifs in temporal networks. Our method is based on the novel notion of egocentric temporal neighborhoods, namely multi-layer structures centered on an ego node. Each temporal layer of the structure consists of the first-order neighborhood of the ego node, and corresponding nodes in sequential layers are connected by an edge. The strength of this approach lies in the possibility of encoding these structures into a unique bit vector, thus bypassing the problem of graph isomorphism in searching for temporal motifs. This allows our algorithm to mine substantially larger motifs with respect to alternative approaches. Furthermore, by bringing the focus on the temporal dynamics of the interactions of a specific node, our model allows to mine temporal motifs which are visibly interpretable. Experiments on a number of complex networks of social interactions confirm the advantage of the proposed approach over alternative non-egocentric solutions. The egocentric procedure is indeed more efficient in revealing similarities and discrepancies among different social environments, independently of the different technologies used to collect data, which instead affect standard non-egocentric measures.

An identification system based on the explicit isomorphism problem

We propose a new identification system based on algorithmic problems related to computing isomorphisms between central simple algebras. We design a statistical zero knowledge protocol which relies on the hardness of computing isomorphisms between orders in division algebras which generalizes a protocol by Hartung and Schnorr, which relies on the hardness of integral equivalence of quadratic forms.

Corrigendum to: V. V. Vasilchikov, “Parallel Algorithm for Solving the Graph Isomorphism Problem”, Modeling and analysis of information systems, vol. 27, no. 1, pp. 86–94, 2020. DOI: https://doi.org/10.18255/1818-1015-2020-1-86-94

In the article by V. V. Vasilchikov “Parallel Algorithm for Solving the Graph Isomorphism Problem” ( Modeling and analysis of information systems, vol. 27, no. 1, pp. 86–94, 2020; DOI: https://doi.org/10.18255/1818-1015-2020-1-86-94) there was a misprint in the layout. In the Table 1, in the last column of the row “Degree of graph” the value should be 3000 (instead of 300). The corrected “Table 1” is shown below. The editors apologise for the inconvenience.

Isomorphism Problem in a Special Class of Banach Function Algebras and its Application

Given a weight function τ, we introduce a new class of Banach function algebras with respect to τ, denoted by C_0b(X, τ ). We provide a complete solution to the isomorphism problem in this class. We further characterize the BSE-extension and the Inoue-Doss ideal associated with it. As an application of our results, we show the equivalence of the four statements: (i) C_0b(X, τ) is of BSE, (ii) C_0b(X, τ) is of BED, (iii) C_0b(X, τ) is Tauberian and (iv) τ is bounded.

On the modular isomorphism problem for groups of class 3 and obelisks

Abstract We study the Modular Isomorphism Problem applying a combination of existing and new techniques. We make use of the small group algebra to give a positive answer for two classes of groups of nilpotency class 3. We also introduce a new approach to derive properties of the lower central series of a finite 𝑝-group from the structure of the associated modular group algebra. Finally, we study the class of so-called 𝑝-obelisks which are highlighted by recent computer-aided investigations of the problem.