8,725 research outputs found
Squarepants in a Tree: Sum of Subtree Clustering and Hyperbolic Pants Decomposition
We provide efficient constant factor approximation algorithms for the
problems of finding a hierarchical clustering of a point set in any metric
space, minimizing the sum of minimimum spanning tree lengths within each
cluster, and in the hyperbolic or Euclidean planes, minimizing the sum of
cluster perimeters. Our algorithms for the hyperbolic and Euclidean planes can
also be used to provide a pants decomposition, that is, a set of disjoint
simple closed curves partitioning the plane minus the input points into subsets
with exactly three boundary components, with approximately minimum total
length. In the Euclidean case, these curves are squares; in the hyperbolic
case, they combine our Euclidean square pants decomposition with our tree
clustering method for general metric spaces.Comment: 22 pages, 14 figures. This version replaces the proof of what is now
Lemma 5.2, as the previous proof was erroneou
The Computational Complexity of Knot and Link Problems
We consider the problem of deciding whether a polygonal knot in 3-dimensional
Euclidean space is unknotted, capable of being continuously deformed without
self-intersection so that it lies in a plane. We show that this problem, {\sc
unknotting problem} is in {\bf NP}. We also consider the problem, {\sc
unknotting problem} of determining whether two or more such polygons can be
split, or continuously deformed without self-intersection so that they occupy
both sides of a plane without intersecting it. We show that it also is in NP.
Finally, we show that the problem of determining the genus of a polygonal knot
(a generalization of the problem of determining whether it is unknotted) is in
{\bf PSPACE}. We also give exponential worst-case running time bounds for
deterministic algorithms to solve each of these problems. These algorithms are
based on the use of normal surfaces and decision procedures due to W. Haken,
with recent extensions by W. Jaco and J. L. Tollefson.Comment: 32 pages, 1 figur
Resolving zero-divisors using Hensel lifting
Algorithms which compute modulo triangular sets must respect the presence of
zero-divisors. We present Hensel lifting as a tool for dealing with them. We
give an application: a modular algorithm for computing GCDs of univariate
polynomials with coefficients modulo a radical triangular set over the
rationals. Our modular algorithm naturally generalizes previous work from
algebraic number theory. We have implemented our algorithm using Maple's RECDEN
package. We compare our implementation with the procedure RegularGcd in the
RegularChains package.Comment: Shorter version to appear in Proceedings of SYNASC 201
Graphs of Edge-Intersecting Non-Splitting Paths in a Tree: Representations of Holes-Part II
Given a tree and a set P of non-trivial simple paths on it, VPT(P) is the VPT
graph (i.e. the vertex intersection graph) of the paths P, and EPT(P) is the
EPT graph (i.e. the edge intersection graph) of P. These graphs have been
extensively studied in the literature. Given two (edge) intersecting paths in a
graph, their split vertices is the set of vertices having degree at least 3 in
their union. A pair of (edge) intersecting paths is termed non-splitting if
they do not have split vertices (namely if their union is a path). We define
the graph ENPT(P) of edge intersecting non-splitting paths of a tree, termed
the ENPT graph, as the graph having a vertex for each path in P, and an edge
between every pair of vertices representing two paths that are both
edge-intersecting and non-splitting. A graph G is an ENPT graph if there is a
tree T and a set of paths P of T such that G=ENPT(P), and we say that is
a representation of G.
Our goal is to characterize the representation of chordless ENPT cycles
(holes). To achieve this goal, we first assume that the EPT graph induced by
the vertices of an ENPT hole is given. In [2] we introduce three assumptions
(P1), (P2), (P3) defined on EPT, ENPT pairs of graphs. In the same study, we
define two problems HamiltonianPairRec, P3-HamiltonianPairRec and characterize
the representations of ENPT holes that satisfy (P1), (P2), (P3).
In this work, we continue our work by relaxing these three assumptions one by
one. We characterize the representations of ENPT holes satisfying (P3) by
providing a polynomial-time algorithm to solve P3-HamiltonianPairRec. We also
show that there does not exist a polynomial-time algorithm to solve
HamiltonianPairRec, unless P=NP
Minimal chordal sense of direction and circulant graphs
A sense of direction is an edge labeling on graphs that follows a globally
consistent scheme and is known to considerably reduce the complexity of several
distributed problems. In this paper, we study a particular instance of sense of
direction, called a chordal sense of direction (CSD). In special, we identify
the class of k-regular graphs that admit a CSD with exactly k labels (a minimal
CSD). We prove that connected graphs in this class are Hamiltonian and that the
class is equivalent to that of circulant graphs, presenting an efficient
(polynomial-time) way of recognizing it when the graphs' degree k is fixed
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