Finding longest paths in hypercubes, snakes and coils

Abstract

Since the problem's formulation by Kautz in 1958 as an error detection tool, diverse applications for long snakes and coils have been found. These include coding theory, electrical engineering, and genetics. Over the years, the problem has been explored by many researchers in different fields using varied approaches, and has taken on additional meaning. The problem has become a benchmark for evaluating search techniques in combinatorially expansive search spaces (NP-complete Optimizations). We present an effective process for searching for long achordal open paths (snakes) and achordal closed paths (coils) in n-dimensional hypercube graphs. Stochastic Beam Search provides the overall structure for the search while graph theory based techniques are used in the computation of a generational fitness value. This novel fitness value is used in guiding the search. We show that our approach is likely to work in all dimensions of the SIB problem and we present new lower bounds for a snake in dimension 11 and coils in dimensions 10, 11, and 12. The best known solutions of the unsolved dimensions of this problem have improved over the years and we are proud to make a contribution to this problem as well as the continued progress in combinatorial search techniques.