A DP Approach to Hamiltonian Path Problem

Dynamic Programming hardly requires introduction. Since the term was in introduced by Richard Bellman in 1940’s it has countless applications and it’s power allows to have fast algorithms in cases where at a blush there seems to be no way to avoid exponential time. Such examples include graph algorithms: Bellman-Ford, Floyd-Warshall as well as option pricing, numerical solutions to HJB equations (”backwards in time”), discrete optimal control policies, knapsack small block size problem [2], Smith-Waterman local sequence alignment algorithm and many others. Hamiltonian path (H-path) in directed and undirected graph is one of Karp’s 21 NP-complete problems [3]. The question it asks is to find cycle or path in a given graph which visits every vertex exactly once. Therefore one can see that H-path is the shortest path which visits all nodes in the graph and thus provides the answer to the transportation problem in question. David Zuckerman [4] showed in 1996 that every one of these 21 problems has a constrained optimization version that is impossible to approximate within any constant factor unless P = NP, by showing that Karp’s approach to reduction generalizes to a specific type of approximability reduction. Significant progress has been made in expanding the class of graphs for which polynomial solution does exist: notably Ashay Dharwadker [5] discovered an algorithm for a broad class of highly connected graphs. The intuition behind the presented approach comes from physical objects that seem to have a way to solve some NP-complete problems such as soap bubbles forming (almost) minimal surfaces and protein folding satisfying hydrophilic-hydrophobic Boolean conditions. Consider a cobweb attached to a tree with one end and being pulled by the other end. One can observe that the lowest (average) strain or highest slack in the cobweb is achieved along the longest sequence of web segments. This is of course a Hamiltonian path in the graph represented by the cobweb. If one wants to find it then it makes sense to implement some incremental scheme which akin to a difference scheme for PDE with boundary condition, computes the strains incrementally starting with one attachment point and moving towards the opposite attachment point.