The property (ii) of the sub-graph searched for in section3.2can also be stated as follows: the sub-graph ofGinduced by the vertices ofZ^ contains an arborescenceA, i.e., a graph that satisfies the following 3 properties (figure3.3):
(1,1) (1,2) (1,3) (1,4)
(2,1) (2,2) (2,3) (2,4)
(3,1) (3,2) (3,3) (3,4)
(4,1) (4,2) (4,3) (4,4)
FIG. 3.2–GraphGassociated with a set of 16 candidate zones represented by a grid of 44 square and identical zones. Two zones are considered adjacent if they share a common side.
Each vertex of the graph corresponds to a zone and vice versa. A vertex is identified by a pair (i,j) whereiis the row index andjis the column index. The double arrow between the two vertices (i,j) and (k,l) represents the arc from (i,j) to (k,l) and the arc from (k,l) to (i,j).
Connectivity 39
(i) Each vertex ofAhas at most one predecessor;
(ii) AincludesZ^1 arcs;
(iii) Adoes not contain circuits.
Example 3.3. Figure3.3 shows an example of a connected sub-graph – from the graph Gin figure3.2–and an associated arborescence.
3.3.1 Case Where No Zone is Mandatory
First of all, we are dealing with the case where none of the candidate zones must be necessarily retained in the reserve. We use the Boolean variables xi which, by con-vention, take the value 1 if and only if vertexi–associated with zonezi–is selected and the Boolean variablesyijwhich by convention take the value 1 if and only if the arcði;jÞ,i.e.,the arc from vertexito vertexj, is retained to form the arborescence.
We also use the non-negative variables tiwhich represent a value assigned to each vertex of the graph. By requiring these values to respect some constraints, we are sure to retain a set of arcs that does not form a circuit. This technique is based on a classic formulation of the travelling salesman’s problem by a mixed-integer linear program with a polynomial number of constraints. We thus obtain a formulation of the problem by program P3.1.
(a) (b)
(1,1) (1,2) (1,3) (1,1) (1,2) (1,3)
(2,2) (2,3) (2,2) (2,3)
(3,2) (3,3) (3,4) (3,2) (3,3) (3,4)
(4,3) (4,3)
FIG. 3.3– (a)GZ^¼ ðZ^;UZ^Þ, the sub-graph of graphG of figure3.2induced by the set of verticesZ^ =fð1;1Þ;ð1;2Þ;ð1;3Þ;ð2;2Þ;ð2;3Þ;ð3;2Þ;ð3;3Þ;ð3;4Þ;ð4;3Þg. (b) A, a partial sub-graph ofGZ^ which is an arborescence; the vertices (1, 2) is the root of this arborescence.
P3:1:
Remember thatU is the set of arcs of the graph associated with the candidate zones. For all i2Z, Adji refers to the set of vertices predecessors of vertex i. In other words, Adji ¼ fj2Z :ðj;iÞ 2Ug. M is a sufficiently large constant (e.g., a value greater than or equal to the number of zones in an optimal reserve). The economic function expresses the cost of the zones selected to form the reserve.
Constraints 3.1.1 express the fact that each species must be protected by at least one zone of the reserve. Given a subset of vertices, Z, and^ xits characteristic vector, a vectoryofRj jU defines an arborescence on the sub-graph induced byZ^ if and only if constraints 3.1.2–3.1.8 are satisfied. Constraints 3.1.2 impose that, if vertexiis not selected, then none of the selected arcs should have this vertex as their initial end. If vertex i is selected, the corresponding constraint is inactive. Constraints 3.1.3 express that, in the case where vertexiis not selected, no arc withias its terminal end can be retained. In the case where vertex iis selected, the corresponding con-straint expresses that at most one arc with ias its terminal end can be retained.
Constraint 3.1.4 expresses that the total number of retained arcs is equal to the number of retained vertices, less 1. Constraints 3.1.5, whereMis a sufficiently large constant, eliminate the possibility that the retained arcs form a circuit. These constraints are similar to those used to eliminate the sub-tours in a classic formu-lation of the travelling salesman’s problem by a mathematical program with a polynomial number of constraints. A positive or zero value ti is assigned to each vertexiof the graph. If the arc (i,j) is retained–yij= 1–thentjmust be greater than or equal toti + 1. Thus, all the selected arcs cannot form a circuit. If the arc (i,j) is not retained – yij= 0 – then the corresponding constraint 3.1.5 is always satisfied provided that the valuestiare less than or equal to M−1.
Connectivity 41
The problem can also be formulated as a slightly different mixed-integer linear program using the Boolean variablesuiwhich are equal to 1 if and only if the vertex iis chosen as the root. This gives program P3.2.
P3:2:
Program P3.2 is obtained by replacing constraints 3.1.3 and 3.1.4 in P3.1 by constraints 3.2.3 and 3.2.4. Constraint 3.2.3 requires to choose the root in one and only one vertex ofZ. Constraints 3.2.4 express that any retained verteximust be the terminal end of one and only one arc unless this vertex has been chosen as root – xi−ui= 0 – in which case it must not be the terminal end of any arc. Note that, according to constraints 3.2.4 and since the quantityP
j2Adji yjiis always positive or null, uican take the value 1 only ifxi= 1.
3.3.2 Case Where at Least One Zone is Mandatory
If the problem data are such that at least one vertex is mandatory – for example, because of constraintsP
i2Zkxi1; k2S–this vertex can be chosen as the root of the arborescence sought without loss of generality, and program P3.3 below, where r refers to this vertex, solves the problem.
P3:3:
Program P3.3 is obtained from program P3.2by replacing constraints 3.2.3 and 3.2.4 with constraints 3.3.3 and 3.3.4. Constraints 3.3.3 express that, for all the selected vertices except the root, one and only one arc must have this vertex as its terminal end. It also expresses that no selected arc should have an unselected vertex as its terminal end. Constraint 3.3.4 expresses that no arcs should arrive on the vertex chosen as root. All other constraints are identical to those of program P3.2.