New Results for the Minimum Weight Triangulation Problem

Given a finite set of points in a plane, a triangulation is a maximal set of non-intersecting line segments connecting the points. The weight of a triangulation is the sum of the Euclidean lengths of its line segments. Given a set of points in a plane, the minimum weight triangulation problem is to find a triangulation whose weight is minimal. No polynomial time algorithm is known to solve this problem, and it is unknown whether the problem is NP-hard. The current best polynomial time approximation algorithm produces a triangulation that can be 0(log n) times the weight of the optimal triangulation. We propose an algorithm that triangulates a set P, of n points in a plane in 0(n-cubed) time and that never does worse than the greedy triangulation. The algorithm produces an optimal triangulation if the points P are the vertices of a convex polygon. The algorithm has the flavor of a heuristic proposed by Lingas and analysis similar to his can be performed for our algorithm also, but experimental results indicate that our algorithm performs much better than the heuristic of Lingas. The results comparing the optimal triangulation with the performance of our algorithm, the heuristic of Lingas, and the greedy algorithm are within 0(1) of an optimal triangulation. We investigate issues of local optimality pertaining to known triangulation algorithms. We define the notion of k-optimality which suggests an interesting new approach to studying triangulation algorithms. We restate the minimum weight triangulation problem as a graph problem and show that NP-hardness of a closely related graph problem. Finally, we show that the constrained problem of computing the minimum weight of triangulation, given a set of points in a plane and enough edges to form a triangulation, is NP-hard. These results are an advance towards a proof that the minimum weight triangulation problem is NP-hard

New Results for the Minimum Weight Triangulation Problem

The current best polynomial time approximation algorithm produces a triangulation that can be O(log n) times the weight of the optimal triangulation. We propose an algorithm that triangulates a set P of n points in a plane in O(n3) time and that never does worse than the greedy triangulation. We investigate issues of local optimality pertaining to known triangulation algorithms and suggest an interesting new approach to studying triangulation algorithms. We restate the minimum weight triangulation problem as a graph problem and show the NP-hardness of a closely related graph problem. Finally, we show that the constrained problem of computing the minimum weight triangulation, given a set of points in a plane and enough edges to form a triangulation, is NP-hard. These results are an advance towards a proof that the minimum weight triangulation problem is NP-hard

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

Potential Maximal Clique Algorithms for Perfect Phylogeny Problems

Kloks, Kratsch, and Spinrad showed how treewidth and minimum-fill, NP-hard combinatorial optimization problems related to minimal triangulations, are broken into subproblems by block subgraphs defined by minimal separators. These ideas were expanded on by Bouchitt\'e and Todinca, who used potential maximal cliques to solve these problems using a dynamic programming approach in time polynomial in the number of minimal separators of a graph. It is known that solutions to the perfect phylogeny problem, maximum compatibility problem, and unique perfect phylogeny problem are characterized by minimal triangulations of the partition intersection graph. In this paper, we show that techniques similar to those proposed by Bouchitt\'e and Todinca can be used to solve the perfect phylogeny problem with missing data, the two- state maximum compatibility problem with missing data, and the unique perfect phylogeny problem with missing data in time polynomial in the number of minimal separators of the partition intersection graph

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

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

Computational Approaches to Lattice Packing and Covering Problems

We describe algorithms which address two classical problems in lattice geometry: the lattice covering and the simultaneous lattice packing-covering problem. Theoretically our algorithms solve the two problems in any fixed dimension d in the sense that they approximate optimal covering lattices and optimal packing-covering lattices within any desired accuracy. Both algorithms involve semidefinite programming and are based on Voronoi's reduction theory for positive definite quadratic forms, which describes all possible Delone triangulations of Z^d. In practice, our implementations reproduce known results in dimensions d <= 5 and in particular solve the two problems in these dimensions. For d = 6 our computations produce new best known covering as well as packing-covering lattices, which are closely related to the lattice (E6)*. For d = 7, 8 our approach leads to new best known covering lattices. Although we use numerical methods, we made some effort to transform numerical evidences into rigorous proofs. We provide rigorous error bounds and prove that some of the new lattices are locally optimal.Comment: (v3) 40 pages, 5 figures, 6 tables, some corrections, accepted in Discrete and Computational Geometry, see also http://fma2.math.uni-magdeburg.de/~latgeo

Computational Geometry Column 42

A compendium of thirty previously published open problems in computational geometry is presented.Comment: 7 pages; 72 reference