6 research outputs found
On a Linear Program for Minimum-Weight Triangulation
Minimum-weight triangulation (MWT) is NP-hard. It has a polynomial-time
constant-factor approximation algorithm, and a variety of effective polynomial-
time heuristics that, for many instances, can find the exact MWT. Linear
programs (LPs) for MWT are well-studied, but previously no connection was known
between any LP and any approximation algorithm or heuristic for MWT. Here we
show the first such connections: for an LP formulation due to Dantzig et al.
(1985): (i) the integrality gap is bounded by a constant; (ii) given any
instance, if the aforementioned heuristics find the MWT, then so does the LP.Comment: To appear in SICOMP. Extended abstract appeared in SODA 201
Minimum-weight triangulation is NP-hard
A triangulation of a planar point set S is a maximal plane straight-line
graph with vertex set S. In the minimum-weight triangulation (MWT) problem, we
are looking for a triangulation of a given point set that minimizes the sum of
the edge lengths. We prove that the decision version of this problem is
NP-hard. We use a reduction from PLANAR-1-IN-3-SAT. The correct working of the
gadgets is established with computer assistance, using dynamic programming on
polygonal faces, as well as the beta-skeleton heuristic to certify that certain
edges belong to the minimum-weight triangulation.Comment: 45 pages (including a technical appendix of 13 pages), 28 figures.
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On a linear program for minimum-weight triangulation
Minimum-weight triangulation (MWT) is NP-hard. It has a polynomial-time constant-factor approximation algorithm, and a variety of effective polynomial-time heuristics that, for many instances, can find the exact MWT. Linear programs (LPs) for MWT are well-studied, but previously no connection was known between any LP and any approximation algorithm or heuristic for MWT. Here we show the first such connections: For an LP formulation due to Dantzig, Hoffman, and Hu [Math. Programming, 31 (1985), pp. 1-14], (i) the integrality gap is constant, and (ii) given any instance, if the aforementioned heuristics find the MWT, then so does the LP. © 2014 Society for Industrial and Applied Mathematics
Studies of several tetrahedralization problems
The main purpose of decomposing an object into simpler components is to simplify a
problem involving the complex object into a number of subproblems having simpler
components. In particular, a tetrahedralization is a partition of the input domain in
R3 into a number of tetrahedra that meet only at shared faces. Tetrahedralizations
have applications in the finite element method, mesh generation, computer graphics,
and robotics.
This thesis investigates four problems in tetrahedralizations and triangulations.
The first problem is on the computational complexity of tetrahedralization detections.
We present an O(nm log n) algorithm to determine whether a set of line segments .C
is the edge set of a tetrahedralization, where m is the number of segments and n is
the number of endpoints in .C. We show that it is NP-complete to decide whether .C
contains the edge set of a tetrahedralization. We also show that it is NP-complete to
decide whether .C is tetrahedralizable. The second problem is on minimal tetrahedralizations.
After deriving some properties of the graph of polyhedra, we identify a class of polyhedra and show that this class of polyhedra can be minimally tetrahedralized
in O(n²) time. The third problem is on the tetrahedralization of two nested convex
polyhedra. We give a method to tetrahedralize the region between two nested convex
polyhedra into a linear number of tetrahedra without introducing Steiner points.
This result answers an open problem raised by Bern [16]. The fourth problem is on
the lower bound for β-skeletons belonging to minimum weight triangulations. We
prove a lower bound on β (β = [one sixth times the square root of two times the square root of 3] + 45 such that if β is less than this value,
the β-skeleton of a point set may not always be a subgraph of the minimum weight
triangulation of this point set. This result settles Keil's conjecture [62]