Related papers: Polynomial braid combing
We introduce the notion of a braid group parametrized by a ring, which is defined by generators and relations and based on the geometric idea of painted braids. We show that the parametrized braid group is isomorphic to the semi-direct…
We prove a characterization of all polynomial-time computable queries on the class of interval graphs by sentences of fixed-point logic with counting. More precisely, it is shown that on the class of unordered interval graphs, any query is…
We consider the disjoint bilinear programming problem in which one of the disjoint subsets has the structure of an acute-angled polytope. An optimality criterion for such a problem is formulated and proved, and based on this, a polynomial…
This note concerns the theoretical algorithmic problem of counting rational points on curves over finite fields. It explicates how the algorithmic scheme introduced by Schoof and generalized by the author yields an algorithm whose running…
We consider several subgroup-related algorithmic questions in groups, modeled after the classic computational lattice problems, and study their computational complexity. We find polynomial time solutions to problems like finding a subgroup…
We use the classical interpretation of the braid group $B_3$ as a central extension of the modular group $\text{PSL}_2\left(\mathbb{Z}\right)$ to establish new and fundamental properties of $B_3$ using the theory of continued fractions. In…
In the Edge Coloring problem, we are given an undirected graph $G$ with $n$ vertices and $m$ edges, and are tasked with finding the smallest positive integer $k$ so that the edges of $G$ can be assigned $k$ colors in such a way that no two…
The Transversal problem, i.e, the enumeration of all the minimal transversals of a hypergraph in output-polynomial time, i.e, in time polynomial in its size and the cumulated size of all its minimal transversals, is a fifty years old open…
The Burau representation of the braid group can be used to recover the Alexander polynomial of the closure of a braid. We define twisted Burau maps and use them to compute twisted Alexander polynomials.
In his initial paper on braids E.Artin gave a presentation with two generators for an arbitrary braid group. We give analogues of this Artin's presentation for various generalizations of braids.
We consider the topological complexity of subgroups of Artin's braid group consisting of braids whose associated permutations lie in some specified subgroup of the symmetric group. We give upper and lower bounds for the topological…
This article presents a strongly polynomial-time algorithm for the general linear programming problem. This algorithm is an implicit reduction procedure that works as follows. Primal and dual problems are combined into a special system of…
We complete the complexity classification by degree of minimizing a polynomial over the integer points in a polyhedron in $\mathbb{R}^2$. Previous work shows that optimizing a quadratic polynomial over the integer points in a polyhedral…
A subset of Q^n is called semilinear (or piecewise linear) if it is Boolean combination of linear half-spaces. We study the computational complexity of the constraint satisfaction problem (CSP) over the rationals when all the constraints…
The notion of a (polynomial) kernelization from parameterized complexity is a well-studied model for efficient preprocessing for hard computational problems. By now, it is quite well understood which parameterized problems do or…
A classical problem in Distance Geometry, with multiple practical applications (in molecular structure determination, sensor network localization etc.) is to find the possible placements of the vertices of a graph with given edge lengths.…
It is shown that for graph groups (right-angled Artin groups) the conjugacy problem as well as a restricted version of the simultaneous conjugacy problem can be solved in polynomial time even if input words are represented in a compressed…
We describe Artin's braid group on a (fixed) finite number of strings as a crossed module over itself. In particular, we interpret the braid relations as crossed module structure relations.
Dehn twists around simple closed curves in oriented surfaces satisfy the braid relations. This gives rise to a group theoretic from the braid group to the mapping class group. We prove here that this map is trivial in stable homology with…
A polyomino is a polygonal region with axis parallel edges and corners of integral coordinates, which may have holes. In this paper, we consider planar tiling and packing problems with polyomino pieces and a polyomino container $P$. We give…