Computing nonsimple polygons of minimum perimeter

We consider the Minimum Perimeter Polygon Problem (MP3): for a given set V of points in the plane, find a polygon P with holes that has vertex set V , such that the total boundary length is smallest possible. The MP3 can be considered a natural geometric generalization of the Traveling Salesman Problem (TSP), which asks for a simple polygon with minimum perimeter. Just like the TSP, the MP3 occurs naturally in the context of curve reconstruction. Even though the closely related problem of finding a minimum cycle cover is polynomially solvable by matching techniques, we prove how the topological structure of a polygon leads to NP-hardness of the MP3. On the positive side, we provide constant-factor approximation algorithms. In addition to algorithms with theoretical worst-case guarantess, we provide practical methods for computing provably optimal solutions for relatively large instances, based on integer programming. An additional difficulty compared to the TSP is the fact that only a subset of subtour constraints is valid, depending not on combinatorics, but on geometry. We overcome this difficulty by establishing and exploiting geometric properties. This allows us to reliably solve a wide range of benchmark instances with up to 600 vertices within reasonable time on a standard machine. We also show that restricting the set of connections between points to edges of the Delaunay triangulation yields results that are on average within 0.5% of the optimum for large classes of benchmark instances.

Si la-Substitution of Alkyl Nitrates: Synthesis, Structural Characterization, and Sensitivity Studies of Highly Explosive (Nitratomethyl)-, Bis(nitratomethyl)-, and Tris(nitratomethyl)silanes and Their Corresponding Carbon Analogues

Evangelisti C, Klapoetke TM, Krumm B, et al. Si la-Substitution of Alkyl Nitrates: Synthesis, Structural Characterization, and Sensitivity Studies of Highly Explosive (Nitratomethyl)-, Bis(nitratomethyl)-, and Tris(nitratomethyl)silanes and Their Corresponding Carbon Analogues. INORGANIC CHEMISTRY. 2010;49(11):4865-4880.A series of analogous nitratomethyl compounds of carbon and silicon of the formula types Me3EICH2ONO2 (1a/1b), Me2EI(CH2ONO2)(2) (2a/2b), MeEI(CH2ONO2)(3) (3a/3b), (CH2)(4)EI(CH2ONO2)(2) (4a/4b), and (CH2)(5)EI(CH2ONO2)(2) (5a/5b) were synthesized [El = C (a), Si (b); (CH2)(4)EI = (sila)cyclopentane-1,1-diy1; (CH2)(5)EI = (sila)cyclohexane-1,1-diy1]. All compounds were characterized by using NMR, IR, and Raman spectroscopy and mass spectrometry. In addition, the crystal structures of Me2C(CH2ONO2)(2) (2a), (CH2)(4)C(CH2ONO2)(2) (4a), Me2Si(CH2ONO2)(2) (2b), and (CH2)(5)Si(CH2ONO2)(2) (5b) were determined by single-crystal X-ray diffraction. The gas-phase structures of the C/Si analogues la and lb were determined by electron diffraction and compared with the results of quantum chemical calculations at different levels of theory. The thermal stabilities of the C/Si pairs la/lb-5a/5b were investigated by using DSC. In addition, their friction and impact sensitivities were measured with standard BAM methods. The extreme sensitivities of the silicon compounds lb 5b compared to those of the corresponding carbon analogues 1a 5a were discussed in terms of the structures of the C/Si analogues and possible geminal Si center dot center dot center dot O interactions