Related papers: The Word and Geodesic Problems in Free Solvable Gr…
We study equations in groups G with unique m-th roots for each positive integer m. A word equation in two letters is an expression of the form w(X,A) = B, where w is a finite word in the alphabet {X,A}. We think of A,B in G as fixed…
A graph is called (generically) rigid in $\mathbb{R}^d$ if, for any choice of sufficiently generic edge lengths, it can be embedded in $\mathbb{R}^d$ in a finite number of distinct ways, modulo rigid transformations. Here we deal with the…
Polygons are a paramount data structure in computational geometry. While the complexity of many algorithms on simple polygons or polygons with holes depends on the size of the input polygon, the intrinsic complexity of the problems these…
The amount of unstructured text-based data is growing every day. Querying, clustering, and classifying this big data requires similarity computations across large sets of documents. Whereas low-complexity similarity metrics are available,…
Motivated by a historical combinatorial problem that resembles the well-known Josephus problem, we investigate circular partition algorithms and formulate problems in deterministic finite automata with practical algorithms. The historical…
We study the complexity of testing if two given matroids are isomorphic. The problem is easily seen to be in $\Sigma_2^p$. In the case of linear matroids, which are represented over polynomially growing fields, we note that the problem is…
We prove that elements of the Hanoi Towers groups $\mathcal{H}_m$ have depth bounded from above by a poly-logarithmic function $O(\log^{m-2} n)$, where $n$ is the length of an element. Therefore the word problem in groups $\mathcal{H}_m$ is…
In this paper we address the decision problem for a fragment of set theory with restricted quantification which extends the language studied in [4] with pair related quantifiers and constructs, in view of possible applications in the field…
In this study, we consider a class of linear matroid interdiction problems, where the feasible sets for the upper-level decision-maker (referred to as a leader) and the lower-level decision-maker (referred to as a follower) are induced by…
De Berg et al. in [SICOMP 2020] gave an algorithmic framework for subexponential algorithms on geometric graphs with tight (up to ETH) running times. This framework is based on dynamic programming on graphs of weighted treewidth resulting…
This paper analyses stable commutator length in groups Z^r * Z^s. We bound scl from above in terms of the reduced wordlength (sharply in the limit) and from below in terms of the answer to an associated subset-sum type problem. Combining…
We prove that Whitehead's algorithm for solving the automorphism problem in a fixed free group $F_k$ has strongly linear time generic-case complexity. This is done by showing that the ``hard'' part of the algorithm terminates in linear time…
Many loss functions in representation learning are invariant under a continuous symmetry transformation. For example, the loss function of word embeddings (Mikolov et al., 2013) remains unchanged if we simultaneously rotate all word and…
We consider the isomorphism problem for groups specified by their multiplication tables. Until recently, the best published bound for the worst-case was achieved by the n^(log_p n + O(1)) generator-enumeration algorithm. In previous work…
Polynomial optimization problems are infinite-dimensional, nonconvex, NP-hard, and are often handled in practice with the moment-sums of squares hierarchy of semidefinite programming bounds. We consider problems where the objective function…
We study the computational complexity of solving mean payoff games. This class of games can be seen as an extension of parity games, and they have similar complexity status: in both cases solving them is in $\textbf{NP} \cap \textbf{coNP}$…
The discrete logarithm problem in a finite group is the basis for many protocols in cryptography. The best general algorithms which solve this problem have time complexity of $\mathcal{O}(\sqrt{N}\log N)$, and a space complexity of…
We prove that the compressed word problem in a group that is hyperbolic relative to a collection of free abelian subgroups is solvable in polynomial time.
Given a system of equations in a "random" finitely generated subgroup of the braid group, we show how to find a small ordered list of elements in the subgroup, which contains a solution to the equations with a significant probability.…
We study the geodesic growth series of the braid group on three strands, B_3 := <a,b|aba = bab>. We show that the set of geodesics of B_3 with respect to the generating set S := {a,b,a^-1,b^-1} is a regular language, and we provide an…