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.

Although advances in computing power have greatly improved computer chess playing, human chess players still rival their computer counter-parts. Computer algorithms typically use a strategy of exhaustive search, which is unlikely to be used by human players. We hypothesized that human chess players recognize higher order properties of the game and use these properties to limit their need for exhaustive move searching. We used graph theoretic modeling to quantitatively determine three possible higher order properties. We then conducted an experiment by using the higher order properties to preselect moves for a typical exhaustive search chess engine. We played the enhanced chess engine against its unenhanced version in six games. The enhanced version won all six games, regardless of color played, suggesting that pre-selection of moves based on higher order properties of the game is indeed a viable strategy.

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.