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

Dorzán, María Gisela

Gagliardi, Edilma Olinda

Hernández Peñalver, Gregorio

Leguizamón, Mario Guillermo

It is known that the Minimum Weight Triangulation problem is NP-hard. Also the complexity of the Minimum Weight Pseudo-Triangulation problem is unknown, yet it is suspected to be also NP-hard. 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 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 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)

Dorzán, M. Gisela

Leguizamón, Guillermo

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Metaheuristic approaches for MWT and MWPT problems

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XIV Spanish Meeting on Computational Geometry, 27–30 June 2011
Metaheuristic approaches for MWT and MWPT
problems
Maria Gisela Dorzán1, Edilma Olinda Gagliardi1, Gregorio Hernández
Peñalver2, Mario Guillermo Leguizamón1
1 Facultad de Ciencias Físico Matemáticas y Naturales, Universidad Nacional de San Luis,
San Luis, Argentina
{mgdorzan,oli,legui}@unsl.edu.ar
2 Facultad de Informática, Universidad Politécnica de Madrid, Madrid, España
gregorio@fi.upm.es
Abstract. It is known that the Minimum Weight Triangulation problem is NP-hard. Also the com-
plexity of the Minimum Weight Pseudo-Triangulation problem is unknown, yet it is suspected to be also
NP-hard. Therefore we focused on the development of approximate algorithms to find high quality trian-
gulations and pseudo-triangulations of minimum weight. In this work we propose two metaheuristics to
solve these problems: Ant Colony Optimization (ACO) and Simulated Annealing (SA). For the experi-
mental 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 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).
Introduction
In Computational Geometry there are many optimization problems that are either NP-
hard or such that no polynomial algorithms are known to solve them. Examples of
these optimization problems are those related to special geometric configurations, such
as triangulations and pseudo-triangulations, which are interesting to investigate due to
their use in many fields of application.
Minimizing the total length has been one of the main optimality criteria for triangula-
tions and pseudo-triangulations. Indeed, the MinimumWeight Triangulation (MWT) and
the Minimum Weight Pseudo-Triangulation (MWPT) minimize the sum of edge lengths,
providing a quality measure for determining how good is a structure. The complexity
of computing a minimum weight triangulation has been one of the most longstanding
open problems in Computational Geometry, introduced by Garey and Johnson in their
open problems list, and various approximation algorithms were proposed over time. It
has been the subject of numerous investigations for identifying criteria to include certain
edges of the MWT. The paper [2] presents the implementations of the LMT heuristic for
computing MWT problems, that can compute the “exact” MWT for well-behaved point
sets very fast. It is also worth noticing that, to our best knowledge, there are no reports
about demonstration of such results. Aichholzer et al. [1] disprove the conjecture that
the LMT-skeleton coincides with the intersection of all locally minimal triangulations,
LMT(P ), being P a polygon in the plane, even for a convex polygon. Later, Mulzer and
Rote proved in 2006 that MWT construction is an NP-hard problem [12].
CRM Documents, vol. 8, Centre de Recerca Matemàtica, Bellaterra (Barcelona), 2011
79
80 Metaheuristic approaches for MWT and MWPT problems
The complexity of the MWPT problem is unknown. However, Levcopoulos and Gud-
mundsson [8] show that a 12-approximation of an MWPT can be computed in O(n3)
time. They give an O(log n ·w(MST )) approximation of an MWPT, in O(n log n) time,
where w(MST ) is the weight of the minimum Euclidean spanning tree, which is a subset
of the obtained structure.
Given the inherent difficulty of the above mentioned problems, the approximate algo-
rithms arise as alternative candidates for MWT and MWPT problems. These algorithms
can obtain approximate solutions to the optimal solutions, and they can be specific for
a particular problem or they can be part of a general applicable strategy in the resolu-
tion of different problems. The metaheuristic methods satisfy these properties. These
algorithms have a simple implementation and they can efficiently find good solutions for
NP-hard optimization problems [11]. In this work we use the Ant Colony Optimization
(ACO) [4] and Simulated Annealing (SA) [3, 9, 10] metaheuristics, and we compare
them with Delaunay Triangulation and Greedy algorithms for triangulations and pseudo-
triangulations. Previous works about approximations on MWT and MWPT problems
using the ACO and SA metaheuristics, were presented in [5, 7]. It is also worth notic-
ing that, to the best knowledge of the authors, there are no reports in the literature of
extensive experimental evaluations using exact algorithms or metaheuristic techniques.
This paper is organized as follows. In the next section, we review some theoretical
aspects of ACO and SA metaheuristics applied to MWT and MWPT problems. Section 2
resumes the experimental evaluation. The last section is devoted to conclusions and future
works.
1 ACO and SA metaheuristics applied to MWT and
MWPT problems
The ACO metaheuristic involves a family of algorithms in which a colony of artificial
ants cooperate in finding good solutions to difficult discrete optimization problems [4].
An artificial ant in an ACO algorithm is a stochastic constructive procedure that in-
crementally builds a solution by adding opportunely defined solution components to a
partial solution under construction. Therefore, the ACO metaheuristic can be applied to
any combinatorial optimization problem for which a constructive graph can be defined.
For details of designs, implementations, and parameter settings of the ACO-MWT and
ACO-MWPT algorithms, see [5, 7].
The SA metaheuristic tries to minimize the limitation of the local search algorithms,
which stops as soon as they find a local extreme. For that, it is allowed to accept
solutions of worse quality than the current solution with a certain probability. This
probability depends on a parameter T , called temperature, which decreases over the
algorithm iterations according to a decrement rule [3]. In regards of the design of the
algorithms, the parameter settings and the details of implementations for SA-MWT and
SA-MWPT algorithms; see [6]. SA2P-MWPT is an improved version of SA-MWPT,
which involves a double pass under certain criteria, considering the best results obtained
in some temperatures.
XIV Spanish Meeting on Computational Geometry, 27–30 June 2011 81
2 Experimental evaluation
To the best knowledge of the authors, there do not exist collections of instances in the
literature for MWT and MWPT problems. Consequently, no benchmarking data are
publicly available that allow us to compare our proposal with some other algorithm
previously studied. According to that, we design an instance generator. Therefore, we
have generated respectively a collection of 10 instances of size 40/80/120/160/200; i.e.,
a total of 50 problem instances. Each instance is called LDn-i where n denotes the size
of the i-instance, with 1 ≤ i ≤ 10. The instance generator uses different functions of
the CGAL Library. The points are randomly generated, uniformly distributed, and, for
each point (x, y), the coordinates x, y are in [0, 1000]. For the purpose of this work, we
assume that there are non-collinear points. The proposed algorithms were implemented
in C language and run on a BACO parallel cluster.
In Table 1 we show the results for ACO-MWT, SA-MWT, Greedy Triangulation
(GT), and Delaunay Triangulation (DT). The best results were obtained with the SA-
MWT algorithm using local retriangulation neighborhood and with different temperature
decrement rules (FSA, VFSA, and geometric decrement with α = 0.8).
Especially for pseudo-triangulations, the Greedy Pseudo-Triangulation (GPT) algo-
rithm builds a pseudo-triangulation starting with one face. This face has the edges ob-
tained by the convex hull of the point set P , i.e., CH(P ). For the solution construction,
the P set is partitioned in faces. This process finishes when all faces are pseudo-triangles
without interior points. A face is divided in two faces when there are interior points,
or is not a pseudo-triangle. Thus, the partition can be done if i) there are at least one
interior point and two points in the border; or ii) there is no interior point, so two points
located on the border are chosen. Such selection is performed by selecting those points
that generate the edges lead to local minimum weight.
ACO-MWT SA-MWT GT DT
LD40-1 5493047 5463745 5477181 5666348
LD40-2 4661242 4659552 4659552 4722381
LD40-3 5502567 5478923 5489487 5663032
LD40-4 5745772 5745772 5751867 6289829
LD80-1 6242505 6220029 6231682 6462038
LD80-2 7605383 7581868 7581868 8081573
LD80-3 5836037 5828344 5845506 6143637
LD80-4 6217040 6147234 6147234 6460311
Table 1. MWT: The best (smallest) weights obtained with the mentioned
algorithms for sets of 40 and 80 points.
Table 2 shows the results according to the smallest weights obtained using the ACO-
MWPT, SA-MWPT, SA2P-MWPT and GPT algorithms. The best results for the
SA-MWPT and SA2P-MWPT algorithms were obtained with edge-flip neighborhood
and with the FSA temperature decrement rule.
ACO algorithms were first used to show that the results from Greedy algorithms and
Delaunay Triangulation can be improved. Other preliminary results seem to indicate that
the SA and SA2P algorithms are more effective techniques for MWT and MWPT. For
more details or an extended version of this work, please refer to the authors.
82 Metaheuristic approaches for MWT and MWPT problems
ACO-MWPT SA-MWPT SA2P-MWPT GPT
LD40-1 6115636 5817042 4181914 5312131
LD40-2 4442710 4778701 3384958 4292347
LD40-3 5684342 6391410 4169242 5794018
LD40-4 5627098 5575409 5047483 6245196
Table 2. MWPT: The best (smallest) weights obtained with the mentioned
algorithms for sets of 40 points.
3 Conclusions
In this work we present the design of approximate algorithms for solving the Minimum
Weight Triangulation and Minimum Weight Pseudo-Triangulation problems. Another
contribution of this research was the creation of a set of instances for the experimental
evaluation, as there are no available instances with special properties for building trian-
gulations and pseudo-triangulations. From this initial experimental phase we obtained
preliminary results that will guide future experimentation. Actually, we are in the phase
of applying a more methodological approach for the experimental design and running the
set of instances for all sets of points mentioned. Since the metaheuristics have proven to
behave very well in solving this class of NP-hard problems, there are several directions
for further research. We intend to use different parameterizations of the ACO and SA
algorithms to adapt and implement other metaheuristics, and to develop hybrid meta-
heuristics to solve the proposed problems.
References
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