Developing the Branch and Bound Method in the Problem of Searching for the Optimal Cyclic Route (Cyclic Rural Postman Problem)

A mathematical model is constructed for the applied problem of the optimization of closed routes, i.e., the rural postman problem. A two-stage method of the branch-and-bound type is proposed, which finds the solution or establishes the unsolvability of the problem. The first stage of the method includes testing the sufficient unsolvability conditions and the vertex-edge transformation procedure. This reduces the solution time at the second stage by the proposed modification of the Little algorithm. This procedure uses (for the first time) the partition of the solution set into disjoint subsets with the help of three branching rules and computation of the corresponding lower-bound estimates of the optimal solution. The proposed method correctly searches for the optimal solution of the Hamiltonian rural postman problem and of the general and Hamiltonian traveling salesman problems.