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.

The Power of the Weisfeiler-Leman Algorithm for Machine Learning with Graphs

In recent years, algorithms and neural architectures based on the Weisfeiler-Leman algorithm, a well-known heuristic for the graph isomorphism problem, emerged as a powerful tool for (supervised) machine learning with graphs and relational data. Here, we give a comprehensive overview of the algorithm's use in a machine learning setting. We discuss the theoretical background, show how to use it for supervised graph- and node classification, discuss recent extensions, and its connection to neural architectures. Moreover, we give an overview of current applications and future directions to stimulate research.

Isomorphism, canonization, and definability for graphs of bounded rank width

We investigate the interplay between the graph isomorphism problem, logical definability, and structural graph theory on a rich family of dense graph classes: graph classes of bounded rank width. We prove that the combinatorial Weisfeiler-Leman algorithm of dimension (3 k + 4) is a complete isomorphism test for the class of all graphs of rank width at most k. A consequence of our result is the first polynomial time canonization algorithm for graphs of bounded rank width. Our second main result addresses an open problem in descriptive complexity theory: we show that fixed-point logic with counting expresses precisely the polynomial time properties of graphs of bounded rank width.

Computational Complexity of Computing Symmetries in Finite-Domain Planning

Symmetry-based pruning is a powerful method for reducing the search effort in finitedomain planning. This method is based on exploiting an automorphism group connected to the ground description of the planning task { these automorphisms are known as structural symmetries. In particular, we are interested in the StructSym problem where the generators of this group are to be computed. It has been observed in practice that the StructSym problem is surprisingly easy to solve. We explain this phenomenon by showing that StructSym is GI-complete, i.e., the graph isomorphism problem is polynomial-time equivalent to it and, consequently, solvable in quasi-polynomial time. This implies that it is solvable substantially faster than most computationally hard problems encountered in AI. We accompany this result by identifying natural restrictions of the planning task and its causal graph that ensure that StructSym can be solved in polynomial time. Given that the StructSym problem is GI-complete and thus solvable quite efficiently, it is interesting to analyse if other symmetries (than those that are encompassed by the StructSym problem) can be computed and/or analysed efficiently, too. To this end, we present a highly negative result: checking whether there exists an automorphism of the state transition graph that maps one state s into another state t is a PSPACE-hard problem and, consequently, at least as hard as the planning problem itself.

ON GENERIC COMPLEXITY OF THE ISOMORPHISM PROBLEM FOR FINITE SEMIGROUPS

Generic-case approach to algorithmic problems was suggested by A. Miasnikov, V. Kapovich, P. Schupp, and V. Shpilrain in 2003. This approach studies behavior of an algorithm on typical (almost all) inputs and ignores the rest of inputs. In this paper, we study the generic complexity of the isomorphism problem for finite semigroups. In this problem, for any two semigroups of the same order, given by their multiplication tables, it is required to determine whether they are isomorphic. V. Zemlyachenko, N. Korneenko, and R. Tyshkevich in 1982 proved that the graph isomorphism problem polynomially reduces to this problem. The graph isomorphism problem is a well-known algorithmic problem that has been actively studied since the 1970s, and for which polynomial algorithms are still unknown. So from a computational point of view the studied problem is no simpler than the graph isomorphism problem. We present a generic polynomial algorithm for the isomorphism problem of finite semigroups. It is based on the characterization of almost all finite semigroups as 3-nilpotent semigroups of a special form, established by D. Kleitman, B. Rothschild, and J. Spencer, as well as the Bollobas polynomial algorithm, which solves the isomorphism problem for almost all strongly sparse graphs.

Graph isomorphism and Gaussian boson sampling

We introduce a connection between a near-term quantum computing device, specifically a Gaussian boson sampler, and the graph isomorphism problem. We propose a scheme where graphs are encoded into quantum states of light, whose properties are then probed with photon-number-resolving detectors. We prove that the probabilities of different photon-detection events in this setup can be combined to give a complete set of graph invariants. Two graphs are isomorphic if and only if their detection probabilities are equivalent. We present additional ways that the measurement probabilities can be combined or coarse-grained to make experimental tests more amenable. We benchmark these methods with numerical simulations on the Titan supercomputer for several graph families: pairs of isospectral nonisomorphic graphs, isospectral regular graphs, and strongly regular graphs.

Проверка изоморфности двух графов и поиск изоморфных подграфов: подход, основанный на анализе максимально протяженных неразветвляющихся путей / Maximal non-branching paths approach to solve (sub)graph isomorphism problem

Предложен подход к решению проблемы проверки изоморфности двух графов исходя из анализа их максимально протяженных неразветвляющихся путей. На его основе предлагается подход и алгоритм решения частного случая задачи поиска в некотором графе A всех подграфов, изоморфных заданному графу B (а именно, поиск только «вписанных» подграфов), а также определяется само понятие «вписанного» подграфа. «Вписанным» подграфом графа A здесь называется такой его подграф, который может быть «приклеен» к другим частям графа A только за счет ребер, инцидентных лишь граничным вершинам его (подграфа) неразветвляющихся путей максимальной длины (при этом граф A может содержать и иные компоненты связности). Решение частного случая задачи поиска «вписанных» подграфов обобщается для поиска в графе A всех подграфов, изоморфных данному графу-образцу B. Для этого вместо рассмотрения их максимально протяженных неразветвляющихся путей необходимо рассматривать все их ребра. Предложенные подход и алгоритм применимы: (1) как для ориентированных, так и для неориентированных графов, (2) для графов, содержащих более одной компоненты связности/ сильной связности, (3) для графов, содержащих кратные (множественные) ребра и петли. ----------- An approach based on maximal non-branching paths analysis to solve graph isomorphism problem is introduced. An algorithm to solve the particular case of the problem of finding in a some graph A all subgraphs that are isomorphic to given graph B is proposed (only “inscribed” subgraphs can be found this way, not all of them). Here we shall name a subgraph of some given graph A as "inscribed" if (1) this subgraph is "glued" to other parts of A only by edges that connected to those vertices of this subgraph that are begin/ end ones of any max-length non-branching path of it, or (2) this subgraph is a separate connected component of the graph A. The proposed algorithm of finding “inscribed” subgraphs is expanded to solve a whole problem of finding all (not only “inscribed”) subgraphs of a graph A that are isomorphic to given template graph B. To do so one should consider all edges of these graphs instead of their max-length non-branching paths. These proposed approach and algorithm may be implemented to: (1) directed or undirected graphs, (2) graphs that have more than one connected components/ strongly connected components, (3) graphs that contain multiple edges and multiple loops.