The Maximum Clique Problem for Permutation Hamming Graphs

This paper explores a new approach to reduce the maximum clique problem associated with permutation Hamming graphs to smaller clique problems. The vertices of a permutation Hamming graph are permutations of n integers and the edges connect pairs of vertices at a Hamming distance greater than or equal to a threshold d. The maximum clique problem for permutation Hamming graphs is a challenging task due to the size, density and regularity of the graphs. However, symmetry properties, which are still partly unexplored, can help to reduce the problems' size and hardness. A property of edge transitivity with respect to automorphisms is proven and leads to a classification for cycle-equivalent edges. This property enables to reduce the full-size clique problem to a set of significantly smaller (and easier to solve) clique problems. The number of reduced problems can be expressed by means of the partition function of integer numbers. Computational experiments confirm that additional knowledge on the automorphism group leads to a more targeted and efficient solving method for the maximum clique problem associated with permutation Hamming graphs.