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

Molecular phylogenetics of the Metazoan clade Lophotrochozoa

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution September 2003DNA sequencing and phylogenetic analyses were conducted to investigate evolutionary relationships between taxa within the metazoan clade Lophotrochozoa. Chapter 1 presents an introduction to phylogenetics of the Metazoa and the clade Lophotrochozoa. Chapter 2 analyzes higher level relationships between the major groups within the phylum Mollusca using sequences of the nuclear ribosomal large-subunit RNA gene (LSD rDNA). Results presented provide the first molecular evidence for a close relationship between the Scaphopoda and Cephalopoda. Phylogenetic trees with this topology were found to have likelihood scores significantly better than those for phylogenies constrained to fit the Diasoma hypothesis grouping Scaphopoda and Bivalvia as sister taxa. Chapter 3 utilizes LSU rDNA sequences to analyze relationships between diverse phyla within the clade Lophotrochozoa. LSU rDNA sequences were found to provide greater resolution than has been provided by previous analyses of the nuclear small-subunit ribosomal RNA gene (SSU rDNA). Analysis ofLSU rDNA sequences recovered the monophyly of several phyla, such as Mollusca and Anelida, whose members are found to be paraphyletic using SSU rDNA sequences alone. Results also suggest that the clade Platyzoa, including rotifers and platyhelminthes, may have arisen within the Lophotrochozoa, rather than as a sister group to lophotrochozoans. Chapter 4 investigates the Hox gene complement of the bryozoan Bugula turrita. Six Hox genes were recovered, including an ortholog of the posterior class gene Post2, which is a synapomorphy for the Lophotrochozoa. The identification of a Post2 ortholog provides evidence of a close relationship between the Bryozoa and other lophotrochozoan phyla.This work was supported by a grant from the National Science Foundation, DEB-0075618 "Genomic approaches to metazoan evolution; lophotrochozoans and Hox genes" to Kenneth Halanych. Bryozoan Hox research was supported by a Doctoral dissertation Improvement Grant from the National Science Foundation, DEB-OI04984 "Phylogenetic inference from bryozoan Hox genes" to Kenneth Halanych and Yale Passamaneck. Additional support was provided by the Woods Hole Oceanographic Institution Education Office