arXiv subjects

Caroline Mattes

Publications and source records attributed to Caroline Mattes.

3 recordsLinked to original sources

arXiv2023

Complexity of Spherical Equations in Finite Groups

In this paper we investigate computational properties of the Diophantine problem for spherical equations in some classes of finite groups. We classify the complexity of different variations of the problem, e.g., when $G$ is fixed and when $G$ is a part of the input. When the group $G$ is constant or given as multiplication table, we show that the problem always can be solved in polynomial time. On the other hand, for the permutation groups $S_n$ (with $n$ part of the input), the problem is NP-complete. The situation for matrix groups is quite involved: while we exhibit sequences of 2-by-2 matrices where the problem is NP-complete, in the full group $GL(2,p)$ ($p$ prime and part of the input) it can be solved in polynomial time. We also find a similar behaviour with subgroups of matrices of arbitrary dimension over a constant ring.

math.GR↗

arXiv2022

Parallel algorithms for power circuits and the word problem of the Baumslag group

Power 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 $(x,y) \mapsto x\cdot 2^y$. 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.

cs.CC↗

arXiv2022

Improved Parallel Algorithms for Baumslag Groups

The Baumslag group had been a candidate for a group with an extremely difficult word problem until Myasnikov, Ushakov, and Won succeeded to show that its word problem can be solved in polynomial time. Their result used the newly developed data structure of power circuits allowing for a non-elementary compression of integers. Later this was extended in two directions: Laun showed that the same applies to the Baumslag groups $G_{1, q}$ for $q \geq 2$ and we established that the word problem of the Baumslag group $G_{1, 2}$ can be solved in $\mathsf{TC}^2$. In this work we refine the operations on reduced power circuits to further improve upon both results. We show that the word problem of the Baumslag groups $G_{p, pq}$ with $|p|,|q| \geq 1$ can be solved in $\mathsf{uTC}^1$. Moreover, we prove that the conjugacy problem in $G_{p, pq}$ is strongly generically in $\mathsf{uTC}^1$ (meaning that for "most" inputs it is in $\mathsf{uTC}^1$). Finally, for every fixed $g \in G_{1, q}$ (case $p=1$) conjugacy to $g$ can be decided in $\mathsf{uTC}^1$ for all inputs. We further show that the word problem of the Baumslag-Solitar groups $BS_{p, pq}$ is in $\mathsf{uAC}^0(F_2)$ if the input word is given in a quite compressed form and so give a complexity result for a special case of the power word problem for these groups.

math.GR↗