Experimental Results on Hamiltonian-Cycle-Finding Algorithms

Frieze [1] introduced a heuristic polynomial-time algorithm, Ham, for finding Hamiltonian cycles in random graphs with high probability. We wanted to see how this algorithm performs in practice, and whether it could be improved by modifying it. For this purpose, we borrowed an idea from an algorithm by Keydar called SemiHam [2]. SemiHam is a modification of Ham that finds Hamiltonian cycles in semi-random graphs with high probability. (The notions of random and semi-random graphs are beyond the scope of this report.) For simplicity, we call our implementations by the borrowed names Ham and SemiHam, even though we don’t follow completely the original theoretical algorithms. We tested our algorithms on knight’s tour problems. By analogy from chess, we define a knight as a piece that moves in a board either ±a cells horizontally and ±b cells vertically, or ±b cells horizontally and ±a cells vertically, for given a and b. Then, given an m×n rectangular board, we ask whether there exists a sequence of moves in which a knight visits each cell of the board exactly once, and then returns to its starting cell. This is the knight’s tour problem. We denote an instance of this problem by listing its parameters in the form m×n–(a, b). For example, the classical knight’s tour problem involves a regular knight and a regular chessboard, and it is denoted 8×8–(1, 2). A knight’s tour problem corresponds in a trivial way to a Hamiltonian cycle problem on a graph: Each cell of the board corresponds to a vertex, and each legal knight move between cells corresponds to an edge. Then a knight’s tour of the board corresponds to a Hamiltonian cycle in the graph. Vandegriend in [3] analyzes the knight’s tour problem in some detail. He gives several existence and non-existence theorems for different parameter values. He also reports on extensive computer experiments, in which he used an exhaustive Hamiltonian cycle algorithm. (Our algorithms, in contrast, are heuristic, as we mentioned). We will compare our results to his.