Abstract
We introduce a novel Conceptual Framework for finding approximations to both Minimum Weight Triangulation (MWT) and optimal Traveling Salesman Problem (TSP) of planar point sets. MWT is a classical problem of Computational Geometry with various applications, whereas TSP is perhaps the most researched problem in Combinatorial Optimization. We provide motivation for our research and introduce the fields of triangulation and polygonization of planar point sets as theoretical bases of our approach, namely, we present the Isoperimetric Inequality principle, measured via Compactness Index, as a key link between our two stated problems. Our experiments show that the proposed framework yields tight approximations for both problems.
Highlights
Traveling Salesman Problem (TSP), whose optimal solution is the minimum-length Hamiltonian Cycle, is the landmark problem in the field of Combinatorial Optimization
A key part of this framework is GCT algorithm we proposed to create a near-optimal TSP based on Isoperimetric Inequality principle applied to simple triangles having points from planar point sets as vertices
We have experimentally confirmed that, on average, GCT is within 0.63% of Minimum weight triangulation (MWT) in 18 TSPLIB instances
Summary
Traveling Salesman Problem (TSP), whose optimal solution is the minimum-length Hamiltonian Cycle, is the landmark problem in the field of Combinatorial Optimization. Minimum weight triangulation (MWT) is defined as the full triangulation of the planar point set with minimal total edge length; it is commonly referred to as the optimal triangulation Both TSP and MWT had been proven to belong to the class of NP-hard problems. But distinct from, the Euclidean TSP is the planar graph TSP which is the focus of our research This is the version of the TSP in which a planar graph G = (V, E) is given, with weights on the edges of E, and one seeks the minimum cost tour which uses only edges in E.
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