Traveling Salesman Problem for a Bidirectional Graph Using Dynamic Programming

Abstract

Traveling salesman problem (TSP) is studied as a combinatorial optimization problem—a problem that attempts to determine an optimal object from a finite set of objects—which is simple to state but difficult to solve. It is a nondeterministic polynomial-time hard problem, hence, exploration on developing algorithms for the TSP has focused on approximate methods above and beyond exact methods. The mission in the TSP is to determine the shortest (optimal) tour when a salesman travels across many cites. A major challenge is that the salesman must be able to minimize entire tour length. The solution to the TSP experiences eclectic applicability in various fields and thus advances the need for an effectual solution. There have been exertions heretofore to provide time efficient solutions (i.e., exact as well as approximate) for the TSP. Dynamic programming is an effective and powerful method that could be used to solve the TSP. Generally, for solving the TSP, a unidirectional path is provided (i.e., whether the salesman travels from city A to B or city B to A) in any input graph, and so, it becomes easier in determining the shortest tour. However, in our study, we have considered a situation where no directions are specified (i.e., the salesman can travel both from city A to B and from city B to A) in an input graph, and for such a graph (i.e., a bidirectional graph), we will determine the shortest tour using dynamic programming.