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

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

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

Metaheuristic approaches for MWT and MWPT problems

It is known that the Minimum Weight Triangulation problem is NP-hard. Also the complexity of Minimum Weight Pseudo-Triangulation problem is unknown, suspecting that it is also a NP-hard problem. Therefore we focused on the development of approximate algorithms to find high quality triangulations and pseudo-triangulations of minimum weight. In this work we propose the use of two metaheuristics to solve these problems: Ant Colony Optimization (ACO) and Simulated Annealing (SA). For the experimental study we have created a set of instances for MWT and MWPT problems since no reference to benchmarks for these problems were found in the literature. Through the experimental evaluation, we assess the applicability of the ACO and SA metaheuristics for MWT and MWPT problems. These results are compared with those obtained from the application of deterministic algorithms for the same problems (Delaunay Triangulation for MWT and a Greedy algorithm respectively for MWT and MWPT).Presentado en el XII Workshop Agentes y Sistemas Inteligentes (WASI)Red de Universidades con Carreras en Informática (RedUNCI

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

Parallel Greedy Triangulation of a Point Set

A greedy triangulation algorithm takes a set of points in the plane and returns a triangulation of the point set. The triangulation is built by adding the smallest line segment between points that does not intersect any line previously in the triangulation. The greedy triangulation is inexpensive computationally and gives an approximation of the minimum-weight triangulation problem, an NP-hard problem, which is computationally expensive. We present serial and parallel implementations of the greedy triangulation using the following approach: once a line is added to the triangulation, all intersecting lines are removed from consideration. This process is repeated until a triangulation is obtained. We present and analyze experimental wall-time data for the serial and parallel implementations. We show that the parallel version has strong and weak scaling properties, and that this algorithm benefits greatly from parallelism

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

Simulated annealing applied to the MWPT problem

The Minimum Weight Pseudo-Triangulation (MWPT) problem is suspected to be NP-hard. We show here how Simulated Annealing (SA) can be applied for obtaining approximate solutions to the optimal ones. To do that, we applied two SA algorithms, the basic version and our extended hybrid version of SA. Through the experimental evaluation and statistical study we assess the applicability and performance of the SA algorithms. The obtained results show the benefits of using the hybrid version of SA to achieve improved and higher quality solutions for the MWPT problem.Proyecto Tecnologías Avanzadas de Bases de Datos (Universidad Nacional de San Luis, Argentina)Laboratorio de Investigación y Desarrollo en Inteligencia Computacional (Universidad Nacional de San Luis, Argentina)European Science FoundationMinisterio de Ciencia e Innovació

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

Minimizing the stabbing number of matchings, trees, and triangulations

The (axis-parallel) stabbing number of a given set of line segments is the maximum number of segments that can be intersected by any one (axis-parallel) line. This paper deals with finding perfect matchings, spanning trees, or triangulations of minimum stabbing number for a given set of points. The complexity of these problems has been a long-standing open question; in fact, it is one of the original 30 outstanding open problems in computational geometry on the list by Demaine, Mitchell, and O'Rourke. The answer we provide is negative for a number of minimum stabbing problems by showing them NP-hard by means of a general proof technique. It implies non-trivial lower bounds on the approximability. On the positive side we propose a cut-based integer programming formulation for minimizing the stabbing number of matchings and spanning trees. We obtain lower bounds (in polynomial time) from the corresponding linear programming relaxations, and show that an optimal fractional solution always contains an edge of at least constant weight. This result constitutes a crucial step towards a constant-factor approximation via an iterated rounding scheme. In computational experiments we demonstrate that our approach allows for actually solving problems with up to several hundred points optimally or near-optimally.Comment: 25 pages, 12 figures, Latex. To appear in "Discrete and Computational Geometry". Previous version (extended abstract) appears in SODA 2004, pp. 430-43