A Heuristic Algorithm for Solving Triangle Packing Problem

Ruimin Wang,Shuai Liu,Xiaozhuo Qi,Yuqiang Luo,Jianqiang Dong

doi:10.1155/2013/686845

Abstract

The research on the triangle packing problem has important theoretic significance, which has broad application prospects in material processing, network resource optimization, and so forth. Generally speaking, the orientation of the triangle should be limited in advance, since the triangle packing problem is NP-hard and has continuous properties. For example, the polygon is not allowed to rotate; then, the approximate solution can be obtained by optimization method. This paper studies the triangle packing problem by a new kind of method. Such concepts as angle region, corner-occupying action, corner-occupying strategy, and edge-conjoining strategy are presented in this paper. In addition, an edge-conjoining and corner-occupying algorithm is designed, which is to obtain an approximate solution. It is demonstrated that the proposed algorithm is highly efficient, and by the time complexity analysis and the analogue experiment result is found.

Highlights

Solving NP-hard problems is always the bottleneck task for computer science and techniques
A edge-conjoining and corner-occupying algorithm that is a quick and approximate method is designed based on the angle region-filling strategy
This paper has special significance for polygon packing problem, network resource optimization, and so forth, since it is a special case of a polygon packing problem; see, for example, [8–11]

Summary

Introduction

Solving NP-hard problems is always the bottleneck task for computer science and techniques. Since triangle in the plane can be continuously translated and rotated, there are infinite numbers of placements which are detrimental to solve the problem. The strategy in [2] is only to allow a polygon to be rotated, but not to be translated, which translates the polygon packing problem to NP-complete problem. There is no a quick and complete solving algorithm, and the heuristic method is still an ordinary method; see, for example, [3–7] and references therein. A edge-conjoining and corner-occupying algorithm that is a quick and approximate method is designed based on the angle region-filling strategy. This paper has special significance for polygon packing problem, network resource optimization, and so forth, since it is a special case of a polygon packing problem; see, for example, [8–11].

Formation of Angle Region

The Classification of an Angle Region and the Corner-Occupying Action

An Algorithm of Edge-Conjoining and Corner-Occupying

Some Illustrative Examples