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

Tetrahedral mesh improvement by shell transformation

Existing flips for tetrahedral meshes simply make a selection from a few possible configurations within a single shell (i.e., a polyhedron that can be filled up with a mesh composed of a set of elements that meet each other at one edge), and their effectiveness is usually confined. A new topological operation for tetrahedral meshes named shell transformation is proposed. Its recursive callings execute a sequence of shell transformations on neighboring shells, acting like composite edge removal transformations. Such topological transformations are able to perform on a much larger element set than that of a single flip, thereby leading the way towards a better local optimum solution. Hence, a new mesh improvement algorithm is developed by combining this recursive scheme with other schemes, including smoothing, point insertion and point suppression. Numerical experiments reveal that the proposed algorithm can well balance some stringent and yet sometimes even conflict requirements of mesh improvement, i.e., resulting in high-quality meshes and reducing computing time at the same time. Therefore, it can be used for mesh quality improvement tasks involving millions of elements, in which it is essential not only to generate high-quality meshes, but also to reduce total computational time for mesh improvement

Dried fruit quality judging on different sweet cherry varieties grown in Hungary

The modern consumer's habit needs new material to enrich biological complete human foods. The mueslis contain more and more part of dried fruit. The organic nutrition prefers fruit, grown in that area, where they will be consumed. Therefore we thought on cherries, because earlier, almost all kind of fruit, including cherries, were dried. First of all we collect all sweet cherry varieties grown in Hungary (by the National List), (Harsányi &. Mády, 2005) than they were dried similar way to the dried plum. After a short storage the samples were judged, by different consumers. The aim was to know, which variety gives the best result, i.e. which variety is the most suitable to gather round for dried fruit. At the judging we used the methods, worked out by the National Institute for Agricultural Quality Control in Hungary. Our work shows orders, which varieties were the best, from different point of view. (e.g. colour, size; firmness of flesh, flavour, and total score). The best varieties by total score will be shown and characterised shortly (Apostol, 1996; Beschreihende Sortenliste Steinobst 1997; Brózik & Kállay, 2000; Tóth, 1997; Harsányi & Mády, 2005; Horváth, 2004; Tomcsányi, 1979)

Dried fruit quality judging on different sweet cherry varieties grown in Hungary

The modern consumer's habit needs new material to enrich biological complete human foods. The mueslis contain more and more part of dried fruit. The organic nutrition prefers fruit, grown in that area, where they will be consumed. Therefore we thought on cherries, because earlier, almost all kind of fruit, including cherries, were dried. First of all we collect all sweet cherry varieties grown in Hungary (by the National List), (Harsányi &. Mády, 2005) than they were dried similar way to the dried plum. After a short storage the samples were judged, by different consumers. The aim was to know, which variety gives the best result, i.e. which variety is the most suitable to gather round for dried fruit. At the judging we used the methods, worked out by the National Institute for Agricultural Quality Control in Hungary. Our work shows orders, which varieties were the best, from different point of view. (e.g. colour, size; firmness of flesh, flavour, and total score). The best varieties by total score will be shown and characterised shortly (Apostol, 1996; Beschreihende Sortenliste Steinobst 1997; Brózik & Kállay, 2000; Tóth, 1997; Harsányi & Mády, 2005; Horváth, 2004; Tomcsányi, 1979)

A biológiai indikátorok információ-tartalmának felhasználása Budapest környezeti viszonyainak értékelésére

Az ipari tevékenység, közlekedés, valamint a különböző fűtőanyagok felhasználása következtében Budapest talajában és a fák leveleiben mind több (és mind nagyobb mennyiségben) toxikus elem mutatható ki. Számos fafaj levele (mint bio-akkumulátor) a terhelés mértékétől függően képes az egyes elemeket nagyobb mennyiségben felhalmozni. Biológiai indikációjuk alapján meghatározható Budapest egyes területeinek immissziós terhelése, ill. az emberi egészségre toxikus elemek előfordulása

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