5. Computational results

This section presents the experimental results obtained with the recursions of above. The algorithms were implemented in ANSI C. The experimental results were obtained in an Intel Core i7, 2.4 GHz, and 8 GB of RAM running under a home PC. The recursions presented in Propositions 4 and 5 were applied to the ANDP(≤1) and the ANDP(≤2), respectively, whereas the recursion presented in Proposition 6 was applied to ANDP(≤2). They were tested using a large test set, by modifying the Steiner Problem in Graphs (SPG) instances from SteinLib [10]. This library contains many problem classes of widely different graph topologies. Most of the problems were extracted from these classes: C, MC, X, PUC, I080, I160, P6E, P6Z, and WRP3. The SPG problems were customized, transforming them into ANDP instances by means of the following changes. For each considered problem:


Moreover, if the resulting topology was unconnected, the problem instance was discarded. Let us notice that since in the ANDP the terminals cannot be used as intermediate nodes (which implies also that edges between pairs of terminals are not allowed), the cost of a SPG optimum is a lower bound for the optimum of the corresponding ANDP. Therefore they are for ANDP(≤k) with k∈ 1:::2.

Table 1 shows the results obtained by applying the recurrences presented in Propositions 4 and 5. In each one of them, the first column contains the names of the original SteinLib classes with the name of the customized instance. The entries from left to right are:


Topological Properties and Dynamic Programming Approach for Designing the Access Network DOI: http://dx.doi.org/10.5772/intechopen.86223


Table 1. 1 2 Results obtained by applying Dynamic Programming to copt and copt.

ð Þ The LB\_GAP <sup>k</sup> SPG is computed as

$$LB\\_GAP^{(k)}\_{SPG} = 100 \times \frac{c^k\_{opt} - LB\_{SPG}}{LB\_{SPG}} \,. \tag{15}$$

Feasible solutions were obtained here only for i080-112, i080-115, and i160-015 with k ¼ 1 because, as can be seen, the cost is finite. The optimal values of the SPG instances (LBSP G) provided lower bounds for the optimal values of the ANDP (therefore to ANDP(≤k) with k≥0), considering that in the ANDP generation process, all the connections between terminal nodes were deleted and further that ANDP's feasible solution space is more restrictive than of SPG. The experimental <sup>1</sup> results obtained for copt have an average gap with respect to the lower bound of 20.72%. Increasing k to 2 (applying the recursion presented in Proposition 5), feasible solutions were obtained for all the testing networks, and the experimental results obtained have an average gap with respect to the lower bound of 7.01%.

It can be proved that (it is out of the scope of this chapter) increasing k, the following inequality is fulfilled:

$$\frac{c\_{opt}^{k-1}}{c\_{opt}^k} \le 1 + \text{floor}\left(\frac{n\_C}{k}\right) \cdot \left(\frac{1}{k + n\_T}\right) \cdot \left(\frac{c\_{max}}{c\_{min}} - 1\right) \tag{16}$$

Table 2 shows the results obtained. Despite the bound was not good in these cases (due the heterogeneity of costs of the lines), it can help us in some cases to answer the following question: how much can be saved with a higher k?


Table 2.

<sup>Þ</sup> Relation between optimal solutions of ANDPð≤1<sup>Þ</sup> and ANDP<sup>ð</sup> <sup>≤</sup><sup>2</sup> .


#### Table 3.

Lower bounds obtained to ANDP<sup>ð</sup> <sup>≤</sup>2<sup>Þ</sup> by applying Dynamic Programming with State-Space Relaxation.

Table 3 shows the results obtained by applying the recursion presented in Proposition 6. As before the first column contains the names of the original SteinLib classes with the name of the customized instance. The entries from left to right are:


Topological Properties and Dynamic Programming Approach for Designing the Access Network DOI: http://dx.doi.org/10.5772/intechopen.86223


ð Þ The LB\_GAP <sup>2</sup> SSR is computed as

$$LB\\_GAP^{(2)}\_{SSR} = 100 \times \frac{c^2\_{opt} - LB^{(2)}\_{SSR}}{LB^{(2)}\_{SSR}}\tag{17}$$

In general, the gaps related to the lower bounds were low. The rito each terminal site and concentrator site were distinct integers chosen from f1; …jST ∪ SCjg. This lower bound can be increased by modifying the state-space through the application of subgradient optimization to ri. As future work, it is possible to incorporate the method for a better choice of ri.

It can be noticed that the execution times of computing global optimal solution costs were much longer than using Dynamic Programming with State-Space Relaxation.
