6. Conclusions

The implementation of the algorithms was tested on a number of different problems with heterogeneous characteristics. In particular, a set of ANDP instances transforming 18 SPG instances extracted from SteinLib was built. The optimal values for the selected SPG instances are lower bound for the corresponding ANDP. The solutions found by the algorithm were, in average, 21% and 7% lower than the mentioned bounds in ANDP(≤1) and ANDP(≤2), respectively. It is reasonable supposing that the gaps related to the global optimum of the ANDP instances be even lower since the feasible solutions of the ANDP that are also feasible solutions of the original SPG, but not reciprocally. In this sense, remember that in any ANDP instance generated, all the edges between pairs of terminal nodes were deleted (because in our ANDP such connections are not allowed) having the additional constraint that the terminal nodes must have degree one in the solution.

Besides, a Dynamic Programming with State-Space Relaxation algorithm was developed which can give a lower bound in polynomial time. The average gaps with respect to the global optimal solution costs were lower than 20%.

Notice that, as expected, the execution times of the proposed algorithms are strongly dependent on the number of sites, edges, and terminal sites. To sum up, as far as the authors are concerned, the results obtained with the recurrences above are very good, considering that computing the global optimal solution of an ANDP(≤2) is a NP-hard problem.

Applied Mathematics
