How to Find Long Paths Efficiently

We study the complexity of finding long paths in directed or undirected graphs, Given a graph G =(V, E) and a number k our algorithm decides within time O(K! · |V| · |E|) for all u,v ɛ V Whether there exists some path of length k firm u to v. The complexity of this algorithm has to be compared with 0(|V|k−1 · |E|) Which is the worst case behaviour of the algorithms described up to now in the literature, We get similar results for the problems of finding a longest path, a cycle of length k or a longest cycle, respectively. Our approach is based on the idea of representing certain families of sets by subfamilies of small cardiality. We also discuss the border lines of this idea.