Apple Carving Algorithm to Approximate Traveling Salesman Problem from Compact Triangulation of Planar Point Sets

Abstract

We propose a modified version of the Convex Hull algorithm for approximating minimum-length Hamiltonian cycle (TSP) in planar point sets. Starting from a full compact triangulation of a point set, our heuristic "carves out" candidate triangles with the minimal Triangle Inequality Measure until all points lie on the outer perimeter of the remaining partial triangulation. The initial candidate list consists of triangles on the convex hull of a given planar point set; the list is updated as triangles are eliminated and new triangles are thereby exposed. We show that the time and space complexity of the "apple carving" algorithm are O(n 2 ) and O(n), respectively. We test our algorithm using a well-known problem subset and demonstrate that our proposed algorithm outperforms nearly all other TSP tour construction heuristics.

Conceptual Framework for Finding Approximations to Minimum Weight Triangulation and Traveling Salesman Problem of Planar Point Sets

“…6 aims to produce a polygon of planar set S by removing exposed triangles with minimum Triangle Inequality Measure from GCT until n−2 triangles remain, since we know there are n−2 simple triangles in any polygonization of planar point set of n points. The applicability of Step 6 step has been confirmed earlier [22]. In this article we focus on viability of Steps 1 and 2.…”

“…4. Compactness Index range for 2D geometric shapes [22] triangulations and polygonizations of choice. Second, within this proposed framework we introduce the algorithm to find approximations to MWT and TSP.…”

“…7. TSP is fully contained within GCT for berlin52 problem set [22] set, and there are n − 2 = 50 gray triangles denoting TSP polygon. This leaves us with exactly n − h = 44 remaining white triangles, representing the Environment.…”