05 Fakultät Informatik, Elektrotechnik und Informationstechnik
Permanent URI for this collectionhttps://elib.uni-stuttgart.de/handle/11682/6
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Item Open Access On the complexity of conjugacy in amalgamated products and HNN extensions(2015) Weiß, Armin; Diekert, Volker (Prof. Dr. rer. nat. habil.)This thesis deals with the conjugacy problem in classes of groups which can be written as HNN extension or amalgamated product. The conjugacy problem is one of the fundamental problems in algorithmic group theory which were introduced by Max Dehn in 1911. It poses the question whether two group elements given as words over a fixed set of generators are conjugate. Thus, it is a generalization of the word problem, which asks whether some input word represents the identity. Both, word and conjugacy problem, are undecidable in general. In this thesis, we consider not only decidability, but also complexity of conjugacy. We consider fundamental groups of finite graphs of groups as defined by Serre - a generalization of both HNN extensions and amalgamated products. Another crucial concept for us are strongly generic algorithms - a formalization of algorithms which work for "most" inputs. The following are our main results: The elements of an HNN extension which cannot be conjugated into the base group form a strongly generic set if and only if both inclusions of the associated subgroup into the base group are not surjective. For amalgamated products we prove an analogous result. Following a construction by Stillwell, we derive some undecidability results for the conjugacy problem in HNN extensions with free (abelian) base groups. Next, we show that conjugacy is decidable if all associated subgroups are cyclic or if the base group is abelian and there is only one stable letter. Moreover, in a fundamental group of a graph of groups with free abelian vertex groups, conjugacy is strongly generically in P. Moreover, we consider the case where all edge groups are finite: If conjugacy can be decided in time T(N) in the vertex groups, then it can be decided in time O(log N * T(N)) in the fundamental group under some reasonable assumptions on T (here, N is the length of the input). We also derive some basic transfer results for circuit complexity in the same class of groups. Furthermore, we examine the conjugacy problem of generalized Baumslag-Solitar groups. Our main results are: the conjugacy problem in solvable Baumslag-Solitar groups is TC0-complete, and in arbitrary generalized Baumslag-Solitar groups it can be decided in LOGDCFL. The uniform conjugacy problem for generalized Baumslag-Solitar groups is hard for EXPSPACE. Finally, we deal with the conjugacy problem in the Baumslag group, an HNN extension of the Baumslag-Solitar group BS12. The Baumslag group has a non-elementary Dehn function, and thus, for a long time, it was considered to have a very hard word problem, until Miaskikov, Ushakov, and Won showed that the word problem, indeed, is in P by introducing a new data structure, the so-called power circuits. We follow their approach and show that the conjugacy problem is strongly generically in P. We conjecture that there is no polynomial time algorithm which works for all inputs, because the divisibility problem in power circuits can be reduced to this conjugacy problem. Also, we prove that the comparison problem in power circuits is complete for P under logspace reductions.Item Open Access Circuit complexity of group theoretic problems(2021) Weiß, Armin; Diekert, Volker (Prof. Dr. rer. nat.)In dieser kumulativen Habilitationsschrift werden sechs Arbeiten zum Thema "Schaltkreiskomplexität von Gruppentheoretischen Problemen" zusammengefasst. An vorderster Stelle steht hierbei das Wortproblem: Gegeben ein Wort über den Erzeugern einer Gruppe, ist die Frage, ob das Wort das Einselement der Gruppe darstellt. Daneben werden noch weitere Probleme, wie das Konjugationsproblem, das Power-Wortproblem (wie das Wortproblem, aber die Eingabe wird in komprimierter Form gegeben) und das Lösen von Gleichungen betrachtet. Die meisten der hier zusammengefassten Arbeiten betrachten die genannten Probleme für spezielle Klassen von Gruppen und klassifizieren deren Komplexität mit Methoden der Schaltkreiskomplexität. Eine Ausnahme bildet die letzte Arbeit zum Thema Gleichungen: hier liegt der Zusammenhang zur Schaltkreiskomplexität darin, dass sich das Erfüllbarkeitsproblem für Gleichungen in endlichen auslösbaren Gruppen ähnlich verhält wie das Erfüllbarkeitsproblem für CC^0 Schaltkreise.Item Open Access Equation satisfiability in solvable groups(2022) Idziak, Paweł; Kawałek, Piotr; Krzaczkowski, Jacek; Weiß, ArminThe study of the complexity of the equation satisfiability problem in finite groups had been initiated by Goldmann and Russell in (Inf. Comput. 178 (1), 253-262, 10 ) where they showed that this problem is in P for nilpotent groups while it is NP -complete for non-solvable groups. Since then, several results have appeared showing that the problem can be solved in polynomial time in certain solvable groups G having a nilpotent normal subgroup H with nilpotent factor G / H . This paper shows that such a normal subgroup must exist in each finite group with equation satisfiability solvable in polynomial time, unless the Exponential Time Hypothesis fails.Item Open Access On the average case of MergeInsertion(2020) Stober, Florian; Weiß, ArminMergeInsertion, also known as the Ford-Johnson algorithm, is a sorting algorithm which, up to today, for many input sizes achieves the best known upper bound on the number of comparisons. Indeed, it gets extremely close to the information-theoretic lower bound. While the worst-case behavior is well understood, only little is known about the average case. This work takes a closer look at the average case behavior. In particular, we establish an upper bound of nlogn-1.4005n+o(n) comparisons. We also give an exact description of the probability distribution of the length of the chain a given element is inserted into and use it to approximate the average number of comparisons numerically. Moreover, we compute the exact average number of comparisons for n up to 148. Furthermore, we experimentally explore the impact of different decision trees for binary insertion. To conclude, we conduct experiments showing that a slightly different insertion order leads to a better average case and we compare the algorithm to Manacher’s combination of merging and MergeInsertion as well as to the recent combined algorithm with (1,2)-Insertionsort by Iwama and Teruyama.Item Open Access Parallel algorithms for power circuits and the word problem of the Baumslag group(2023) Mattes, Caroline; Weiß, ArminPower circuits have been introduced in 2012 by Myasnikov, Ushakov and Won as a data structure for non-elementarily compressed integers supporting the arithmetic operations addition and (x,y)↦x·2y. The same authors applied power circuits to give a polynomial time solution to the word problem of the Baumslag group, which has a non-elementary Dehn function. In this work, we examine power circuits and the word problem of the Baumslag group under parallel complexity aspects. In particular, we establish that the word problem of the Baumslag group can be solved in NC -even though one of the essential steps is to compare two integers given by power circuits and this, in general, is shown to be P -complete. The key observation is that the depth of the occurring power circuits is logarithmic and such power circuits can be compared in NC .Item Open Access The power word problem in graph products(2024) Lohrey, Markus; Stober, Florian; Weiß, ArminThe power word problem for a group Gasks whether an expression u1x1⋯unxn, where the uiare words over a finite set of generators of Gand the xibinary encoded integers, is equal to the identity of G. It is a restriction of the compressed word problem, where the input word is represented by a straight-line program (i.e., an algebraic circuit over G). We start by showing some easy results concerning the power word problem. In particular, the power word problem for a group Gis uNC1-many-one reducible to the power word problem for a finite-index subgroup of G. For our main result, we consider graph products of groups that do not have elements of order two. We show that the power word problem in a fixed such graph product is AC0-Turing-reducible to the word problem for the free group F2and the power word problems of the base groups. Furthermore, we look into the uniform power word problem in a graph product, where the dependence graph and the base groups are part of the input. Given a class of finitely generated groups Cwithout order two elements, the uniform power word problem in a graph product can be solved in AC0[C=LUPowWP(C)], where UPowWP(C)denotes the uniform power word problem for groups from the class C. As a consequence of our results, the uniform knapsack problem in right-angled Artin groups is NP-complete. The present paper is a combination of the two conference papers (Lohrey and Weiß 2019b, Stober and Weiß 2022a). In Stober and Weiß (2022a) our results on graph products were wrongly stated without the additional assumption that the base groups do not have elements of order two. In the present work we correct this mistake. While we strongly conjecture that the result as stated in Stober and Weiß (2022a) is true, our proof relies on this additional assumption.