Graph Isomorphism in Quasipolynomial Time Parameterized by Treewidth

We extend Babai's quasipolynomial-time graph isomorphism test (STOC 2016) and develop a quasipolynomial-time algorithm for the multiple-coset isomorphism problem. The algorithm for the multiple-coset isomorphism problem allows to exploit graph decompositions of the given input graphs within Babai's group-theoretic framework. We use it to develop a graph isomorphism test that runs in time $n^{\operatorname{polylog}(k)}$ where $n$ is the number of vertices and $k$ is the minimum treewidth of the given graphs and $\operatorname{polylog}(k)$ is some polynomial in $\operatorname{log}(k)$. Our result generalizes Babai's quasipolynomial-time graph isomorphism test.Comment: 52 pages, 1 figur

Presentations and Structural Properties of Self-similar Groups and Groups without Free Sub-semigroups

This dissertation is devoted to the study of self-similar groups and related topics. It consists of three parts. The first part is devoted to the study of examples of finitely generated amenable groups for which every finitely presented cover contains non-abelian free subgroups. The study of these examples was motivated by natural questions about finiteness properties of finitely generated groups. We show that many examples of amenable self-similar groups studied in the literature cannot be covered by finitely presented amenable groups. We investigate the class of contracting self-similar groups from this perspective and formulate a general result which is used to detect this property. As an application we show that almost all known examples of groups of intermediate growth cannot be covered by finitely presented amenable groups. The latter is related to the problem of the existence of finitely presented groups of intermediate growth. The second part focuses on the study of one important example of a self-similar group called the first Grigorchuk group G, from the viewpoint of pro finite groups. We investigate finite quotients of this group related to presentations and group (co)homology. As an outcome of this investigation we prove that the pro finite completion G_hat of this group is not finitely presented as a pro finite group. The last part focuses on a class of recursive group presentations known as L-presentations, which appear in the study of self-similar groups. We investigate the relation of such presentations with the normal subgroup structure of finitely presented groups and show that normal subgroups with finite cyclic quotient of finitely presented groups have such presentations. We apply this result to finitely presented indicable groups without free sub-semigroups

Combinatorial approaches to the group isomorphism problem

The isomorphism problem of finite groups, that is, the task of deciding whether two given finite groups are isomorphic, is one of the most fundamental problems in computational group theory for which we currently do not have efficient algorithmic tools. This is equally true in practical applications, as well as in terms of computational complexity: in the general case, apart from minor improvements, we are essentially stuck with an upper bound of n^O(log n) (obtained from enumerating all (log n)-sized generating sets), where n is the group order. On the other hand, there are currently no substantial lower bounds. In this thesis, we develop new algorithmic perspectives on the group isomorphism problem. We define and analyze a series of combinatorial algorithms in the context of finite groups, and in fact arbitrary relational structures. More precisely, we study the k-dimensional WL-algorithm} (k-WL) for natural numbers k, which is an essential tool for the graph isomorphism problem. It is a crucial subroutine in all state-of-the-art graph isomorphism solvers, and it forms an important building block in Babai’s break-through quasi-polynomial time (n^O((log n)^c)) algorithm for graph isomorphism. It is a combinatorial algorithm with a runtime of n^O(k), that assigns canonical colorings to graphs. It thereby serves as a non-isomorphism test, with important connections to logic, games, and graph structure theory. Our first contribution is the generalization of the WL-algorithm from graphs to relational structures, in terms of three potentially different versions of the WL. We compare these versions, showing that they can be placed in a hierarchy of distinguishing powers. The general result that we prove is that each version is natural under a certain point of view (and can be characterized by a corresponding logic), but asymptotically, it does not matter which version of WL we work with. In particular, we obtain an asymptotically robust notion of the Weisfeiler-Leman dimension for relational structures, which denotes the smallest natural number k, such that the k-dimensional WL-algorithm identifies a given structure up to isomorphism. This allows us to subsequently initiate a descriptive complexity theory of finite groups, where we propose the Weisfeiler-Leman dimension as a natural measure of complexity. We construct a compendium of structural properties and group theoretic constructions that are detectable via a low-dimensional Weisfeiler-Leman algorithm. This includes various major building blocks of group theory, for example, we show that groups share the same multiset of composition factors if they are indistinguishable via 5-WL. We also provide a framework that allows one to easily extend and adapt our results to other group theoretic properties. We thereby uncover far-reaching connections between the WL-dimension and the structure of a finite group, and we provide an effective tool-kit to analyze the WL-algorithm on groups and related algebraic structures. We then employ these tools to derive upper bounds on the WL-dimension of several important group classes. For instance, we show that the WL-dimension of coprime extensions of abelian groups and the WL-dimension of semisimple groups are both bounded by O(log log n). We also identify several natural group classes of bounded WL-dimension. Finally, we discuss lower bounds in two ways: first, we provide explicit examples that certify Weisfeiler-Leman indistinguishability for small dimensions, and second, we devise combinatorial reductions that asymptotically preserve the WL-dimension. The latter provides potential sources for groups of unbounded Weisfeiler-Leman dimension

Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic

On the Combinatorial and Logical Complexities of Algebraic Structures

In this thesis, we investigate the combinatorial and logical complexities of several algebraic structures, including groups, quasigroups, certain families of strongly regular graphs, and relation algebras. In Chapter 3, we leverage the Weisfeiler&ndash;Leman algorithm for groups (Brachter &amp; Schweitzer, LICS 2020) to improve the parallel complexity of isomorphism testing for several families of groups including (i) coprime extensions H ⋉ N where H is O(1)-generated and N is Abelian (c.f., Qiao, Sarma, &amp; Tang, STACS 2011), (ii) direct product decompositions, and (iii) groups without Abelian normal subgroups (c.f., Babai, Codenotti, &amp; Qiao, ICALP 2012). Furthermore, we show that the weaker count-free Weisfeiler&ndash;Leman algorithm is unable to even identify Abelian groups. As a consequence, we obtain that FO fails to capture all polynomial-time computable queries even on Abelian groups. Nonetheless, we leverage the count-free variant of Weisfeiler&ndash; Leman in tandem with bounded non-determinism and limited counting to obtain a new upper bound of &beta;1MAC0 (FOLL) for isomorphism testing of Abelian groups. This improves upon the previous TC0 (FOLL) upper bound due to Chattopadhyay, Toran, &amp; Wagner (ACM Trans. Comput. Theory, 2013). Weisfeiler&ndash;Leman is equivalent to the first in a hierarchy of Ehrenfeucht&ndash;Fra&uml;ıss&acute;e pebble games (Hella, Ann. Pur. Appl. Log., 1989). In Chapter 4, we explore the descriptive complexity theory of finite groups by examining the power of the second Ehrenfeucht-Fra&uml;ıss&acute;e bijective pebble game in Hella&rsquo;s (Ann. Pure Appl. Log., 1989) hierarchy. This is a Spoiler-Duplicator game in which Spoiler can place up to two pebbles each round. While it trivially solves graph isomorphism, it may be nontrivial for finite groups, and other ternary relational structures. We first provide a novel generalization of Weisfeiler-Leman (WL) coloring, which we call 2-ary WL. We then show that the 2-ary WL is equivalent to the second Ehrenfeucht-Fra&uml;ıss&acute;e bijective pebble game in Hella&rsquo;s hierarchy.&nbsp; Our main result is that, in the pebble game characterization, only O(1) pebbles and O(1) rounds are sufficient to identify all groups without Abelian normal subgroups. In particular, we show that within the first few rounds, Spoiler can force Duplicator to select an isomorphism between two such groups at each subsequent round. By Hella&rsquo;s results (ibid.), this is equivalent to saying that these groups are identified by formulas in first-order logic with generalized 2-ary quantifiers, using only O(1) variables and O(1) quantifier depth.&nbsp; In Chapter 5, we show that Graph Isomorphism (GI) is not AC0 -reducible to several problems, including the Latin Square Isotopy problem and isomorphism testing of several families of Steiner designs. As a corollary, we obtain that GI is not AC0 -reducible to isomorphism testing of Latin square graphs and strongly regular graphs arising from special cases of Steiner 2-designs. We accomplish this by showing that the generator-enumeration technique for each of these problems can be implemented in &beta;2FOLL, which cannot compute Parity (Chattopadhyay, Tor&acute;an, &amp; Wagner, ibid.). Finally, in Chapter 6, we shed new light on the spectrum of the relation algebra we call An, which is obtained by splitting the non-flexible diversity atom of 67 into n symmetric atoms. Precisely, we show that the minimum value in Spec(An) is at most 2n6+o(1), which is the first polynomial bound and improves upon the previous bound due to Dodd &amp; Hirsch (J. Relat. Methods Comput. Sci. 2013). We also improve the lower bound to 2n2 + Ω(n&radic;logn). Prior to the work in this thesis, only the trivial bound of n2 + 2n + 3 was known.</p

Proceedings of Sixth International Workshop on Unification

Swiss National Science Foundation; Austrian Federal Ministry of Science and Research; Deutsche Forschungsgemeinschaft (SFB 314); Christ Church, Oxford; Oxford University Computing Laborator

Unsolved Problems in Group Theory. The Kourovka Notebook

This is a collection of open problems in group theory proposed by hundreds of mathematicians from all over the world. It has been published every 2-4 years in Novosibirsk since 1965. This is the 19th edition, which contains 111 new problems and a number of comments on about 1000 problems from the previous editions.Comment: A few new solutions and references have been added or update