The shortest combination of paths that starts and ends with a single node and covers all points in an undirected graph

Question

I need an algorithm(k, s) where

• k is the number of paths
• s is the starting and ending node

and given n number of nodes in an undirected graph in which all nodes are linked to each other, returns k paths to traverse all nodes of which the sum of distances covered by the k paths is the shortest.

Eg Given n = 10 , algorithm(2,5) might give me an array of two arrays such that the sum of the distances covered by the two paths are the shortest and that all nodes are traversed.

[[5,1,2,3,10,5],[5,4,6,7,8,9,5]]

Djikstra's algorithm finds the shortest path from one node to another, but not the shortest combination of k paths.

Yen's algorithm finds k number of shortest paths from one node to another, but not the shortest combination of k paths.

What algorithm can help me find the shortest combination of k paths that starts and end with node s such that all n nodes are covered?

Answer 1

What you are describing above, is the classical Traveling Sales Man problem, many optimization techniques. One such is Ant Colony Optimization .

There are many libraries that implement ACO, you could find one for Ruby, as it stands, there are no easy solutions to Traveling Salesman Problem with classical (mathematical) approach.