Abstract. The paper focuses on finding of the optimal Hamiltonian cycle, when it is regarded with respect to cost, time, distance or difficulty level of the route. The problem is strictly related to the traveling salesman problem proved to be NP-complete for general graphs. The paper gives a heuristic algorithm for finding the optimal spanning cycle in a weighted graph. Its idea is based on optimization of weight losses and reduction the complexity of a problem by reduction the dimension of the graph payoff matrix.&nbsp