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. This revision contains a few improvements in the expositio

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

Neuronal Deletion of Caspase 8 Protects against Brain Injury in Mouse Models of Controlled Cortical Impact and Kainic Acid-Induced Excitotoxicity

system. mice demonstrated superior survival, reduced seizure severity, less apoptosis, and reduced caspase 3 processing. Uninjured aged knockout mice showed improved learning and memory, implicating a possible role for caspase 8 in cognitive decline with aging.Neuron-specific deletion of caspase 8 reduces brain damage and improves post-traumatic functional outcomes, suggesting an important role for this caspase in pathophysiology of acute brain trauma

A Package for Triangulations

Introduction Triangulations of a point set have been intensively studied in computational geometry in these years. Triangulations are very important not only from the viewpoint of their applications to computer graphics, numerical analysis but also from a fact that triangulations are yet another fundamental structure in discrete geometry like hyperplane arrangements. Our group has developed many systems concerning triangulations. Specifically, ffl minimum weight triangulation of a planar point set ffl enumerating all triangulations of a planar point set ffl uniform generation of triangulations of a planar point set ffl enumeration of all regular triangulations of a point set in general dimensions In this video, we demonstrate our results of the abovementioned problems, mainly using animation. 2 Minimum weight triangulation Kyoda [2] presents a branch-and-cut algorithm for computing the minim

The Minimum Weight Triangulation Problem With Few Inner Points

We propose to look at the computational complexity of 2-dimensional geometric optimization problems on a finite point set with respect to the number of inner points (that is, points in the interior of the convex hull). As a case study, we consider the minimum weight triangulation problem. Finding a minimum weight triangulation for a set of n points in the plane is not known to be NP-hard nor solvable in polynomial time, but when the points are in convex position, the problem can be solved in O(n ) time by dynamic programming. We extend the dynamic programming approach to the general problem and describe an exact algorithm which runs in O(6 log n) time where n is the total number of input points and k is the number of inner points. If k is taken as a parameter, this is a fixed-parameter algorithm. It also shows that the problem can be solved in polynomial time if k = O(log n). In fact, the algorithm works not only for convex polygons, but also for simple polygons with k interior points