3. Access Network Design Problem

The Access Network Design Problem is defined as follows.

Definition 2 (ANDP—Access Network Design Problem). Let GA ¼ ðS; E<sup>1</sup> ∪ E2Þ be the graph of feasible connections on the access network and C the matrix of connection costs defined previously. The Access Network Design Problem ðS; E<sup>1</sup> ∪ E2;CÞ consists in finding a subgraph of GA of minimum cost such that ∀i ∈ST; there exists a path from i to some site j∈SD of the backbone network.

Notation 1. ΓANDP denotes the space of feasible solutions of ANDPðS; E<sup>1</sup> ∪ E2;CÞ that do not have any cycle and with an output only toward the backbone network ∀t∈ST. These have forest topology as we illustrate in Figure 2.

In order to define these problems in terms of graph theory, the following notation is introduced:


Figure 2. A feasible solution of ANDP.


The General Access Network Design Problem (GANDP) consists of finding a minimum-cost subgraph H ⊂ G such that all the sites of ST are communicated with some node of the backbone. This connection can be direct or through intermediate concentrators. The use of terminal sites as intermediate nodes is not allowed; this implies that they must have degree one in the solution.

The GANDP is here simplified by collapsing the backbone into a fictitious node and given the name of "Access Network Design Problem." The equivalence between both problems, GANDP and ANDP, as well as the NP-hardness of the ANDP, is proved in [7].

This work concentrates on the ANDP with the objective of proposing a new approach for solving this problem. We study different results related to the topological structure of the ANDP solutions. In particular we present results that characterize the topologies of the feasible solutions of an ANDP instance. The following proposition shows the topological form of the feasible solutions of ΓANDP for a given ANDP instance.

Proposition 1. Given an ANDP with associated graph GA ¼ ðS; E<sup>1</sup> ∪ E2Þ and ˛ ˝ matrix of connection costs <sup>C</sup>. If the subnetwork <sup>H</sup> <sup>¼</sup> ST <sup>∪</sup> <sup>S</sup>; <sup>E</sup> (with <sup>S</sup>⊆SC <sup>∪</sup> SD and E⊆E<sup>1</sup> ∪ E2) is an optimal solution of ΓANDP, it is composed of a set of disjoint trees H ¼ fH1; …; Hmg that satisfy:

$$\mathbf{1. \forall H\_l \in H, \ \exists j \in \mathbf{S}\_D \text{ } unique \ / j \in H\_l.$$

$$\text{2. } \forall H\_l \in H, \exists \ a \ subset \ \mathcal{S}\_T^l \subset \mathcal{S}\_T, \mathcal{S}\_T^l \neq \mathcal{Q} \ \_{\overline{\mathcal{S}\_T}} \mathsf{T} \\ \text{NODES}(H\_l)$$

3.⋃<sup>m</sup> <sup>l</sup>¼<sup>1</sup>S<sup>l</sup> <sup>T</sup> ¼ ST

Proof. Trivial.

The following propositions present results that characterize the structure of the global optimal solution.

Proposition 2. Let ANDP ðS; E<sup>1</sup> ∪ E2;CÞ be a problem where sc ∈ SC, s∈ SC ∪ SD and s∈ ST ∪ SC such that fðs; scÞ;ðsc; sÞg⊂E<sup>1</sup> ∪ E<sup>2</sup> and ∃sw ∈SD=cs,sw <cs,sc þ csc,s: Then, if TA ∈ ΓANDP is a globally optimal solution, it is fulfilled that g sð Þ<sup>c</sup> ≥3 in TA, ∀sc ∈ TA, sc ∈SC.

Proof. Let us suppose that there exists TA ∈ ΓANDP global optimal solution such that ∃sc ∈TA a concentrator site with g <3 in TA. If g sð <sup>c</sup>Þ ¼ 1; then sc is a sc pendant in TA; therefore, eliminating this, a feasible solution of smaller cost would be obtained. This is a contradiction; hence, g sð Þ<sup>c</sup> ¼6 1. If g sð <sup>c</sup>Þ ¼ 2, let s∈ SC ∪ SD be the site adjacent to sc in TA which its output site is toward the backbone network. Let s∈ST ∪ SC be the other adjacent site in TA. Considering the network H ¼ ðTAf g sc Þ ∪ fð < csc,s s; swÞg, where sw ∈SD satisfies cs,sw , it is fulfilled:

$$\text{COST}(H) = \text{COST}(T\_A) - c\_{\mathfrak{s},\mathfrak{s}\_\mathfrak{s}} - c\_{\mathfrak{s}\_\mathfrak{s},\mathfrak{s}} + c\_{\mathfrak{s},\mathfrak{s}\_\mathfrak{w}} < \text{COST}(T\_A) \tag{1}$$

Furthermore, it is easy to see that H ∈ ΓANDP. Hence, this implies that H is a better feasible solution compared with TA. This is a contradiction, entailing that g sð Þ<sup>c</sup> ≥3 in TA, as required and completing the proof.

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

Proposition 3. Given an ANDP ðS; E<sup>1</sup> ∪ E2;CÞ such that for any three sites ðs1; s2; s3Þ, with s<sup>1</sup> ∈ST ∪ SC, s<sup>2</sup> ∈SC and s<sup>3</sup> ∈SC ∪ SD, the strict triangular inequality is satisfied, i.e., cs1,sk < csi,Sj þ csj,sk , i, j, k∈f1; 2; 3g. Then, if TA ∈ ΓANDP is a globally optimal solution, it is fulfilled that g sð Þ<sup>c</sup> ≥3 in TA, ∀sc ∈TA, sc ∈SC.

Proof. As in the previous proposition, let us suppose that there exists TA ∈ ΓANDP global optimal solution such that ∃sc ∈TA, a concentrator site with g sð Þ<sup>c</sup> <3 in TA. Clearly g sð Þ<sup>s</sup> must be different to 1. Now, let us consider the case g sð Þ¼ <sup>c</sup> 2 inTA. Let s1, s<sup>2</sup> be the adjacent sites to sc in TA. By hypothesis cs1,s<sup>2</sup> <cs1,sc þ csc,s<sup>2</sup> . Considering the network TA ¼ ðTAf g sc Þ ∪ fðs1; s2Þg, a feasible solution is found, and moreover

$$\text{COST}(\overline{T}\_A) = \text{COST}(T\_A) - \mathfrak{c}\_{\mathfrak{s}\_1,\mathfrak{s}\_\varepsilon} - \mathfrak{c}\_{\mathfrak{s}\_\varepsilon,\mathfrak{s}\_2} + \mathfrak{c}\_{\mathfrak{s}\_1,\mathfrak{s}\_2} < \text{COST}(T\_A) \tag{2}$$

This is a contradiction; therefore, g sð Þ<sup>c</sup> ≥3 in TA, hence completing the proof.

The next section presents algorithms applied to the ANDP(≤k) with <sup>k</sup><sup>∈</sup> <sup>f</sup>1; <sup>2</sup>g. A way of computing the global optimal solution cost of it using the Dynamic Programming approach is obtained. Considering that the ANDP(≤1) is a NP-hard problem, we obtain lower bounds to the global optimal solution cost by Dynamic Programming with State-Space Relaxation in polynomial time.
