Additional constraints

Although the constraints above are sufficient to model the gfrp, we may want to introduce additional constraints in order to improve the strength of the model and boost its running time. In Sections 6.5.1 and 6.5.2, we describe two of these constraints, which may simply be added to the model without altering the set of optimal solutions. On the other hand, in Section 6.5.3, we describe an additional constraint that, while it may sometimes discard a small set of valid solutions for the original problem, it often either still leads to optimal solutions or reaches solutions of nearly optimal value for the initial problem.

In Section 6.7, we further discuss the benefits of the latter constraints and analyze the resulting improvement in running time.

6.5.1 Maximum deadline constraints

Let C be any subset of E^−→_B, and let δmax= maxi∈Cδi. From Constraints (6.7) it is clear that we cannot have ϑ_ij > (δ_max− σ_i) for any i ∈ C. In particular, let C^∗ ⊆ C be a set of barriers that can be part of a viable solution, and let i be the last constructed barrier of this set. Clearly, P

j∈C^∗σ_j ≤ δ_max. We can generalize this idea for C itself through the following new constraint:

X

i∈C

σ_iy_i ≤ δ_max (6.14)

Of course this constraint is interesting only when P

i∈Cσ_i > δ_max. Although there is an exponential number of such constraints, we will add the following O(|E^−→_B|) constraints set up by Algorithm 2 and prove that these constraints dominate all others.

Algorithm 2 Set of maximum deadline constraints.

1: ξ ← barriers in E^−→_B sorted in increasing order of deadline.

2: for i ← 1 to |ξ| do

Proposition 2. The set of constraints set up by Algorithm 2 dominates all Constraints (6.14).

Proof. Let us assume, without loss of generality, that all deadlines are distinct. Consider the i^th iteration of the loop on Line 2 and let C = {ξ_1:i}. Also, let C⁰ ⊆ E^−→_B such that max_k∈C⁰{δ_k} ≤ δ_ξi and C⁰ 6= C. As we can only add to C⁰ barriers with deadline less than or equal to δ_max= δ_ξi, C is maximal in this regard, and consequently C⁰ ⊂ C. Therefore, the constraint for when δ_max = δ_ξi added to the model by Algorithm 2 dominates any other Constraint (6.14) with right-hand side δξi. Since this is valid for all deadlines, the propositions holds.

6.5.2 Clique constraints

Another set of relevant constraints is the one composed by clique constraints. A clique constraint is employed when there are subsets of binary variables that are mutually exclusive. In our setting, it means finding a subset of barriers that cannot be constructed in the same solution. As an illustration, consider Figure 6.7. There, i and j are the two barriers we want to determine whether they can figure in the same solution. The paths τ_ij = ρ^∗_i + ρ_ij + σ_j and τ_ji = ρ^∗_j + ρ_ji+ σ_i are, respectively, the shortest paths starting on the boundary and connecting i and j, with the construction of i preceding the construction of j and vice-versa. It is clear that if we have |τij| > δ_j and |τji| > δ_i then we cannot have i and j in the same solution as there is no solution where one can precede the other. In this case, we can add constraint y_i+ y_j ≤ 1 to the model.

6.5.3 k-gfrp

It is often interesting to restrict hard problems in order to make them easier to solve, at least in practice, while still attaining solutions close to the optimum for the unrestricted problem. For the gfrp one may want to restrict the in-degrees and out-degrees of the vertices that are endpoints of barriers constructed in an optimal solution. With this in mind, we define a gfrp variant called k-gfrp that restricts the in-degree and out-degree of the vertices not on ∂P to be less than or equal to k in optimal solutions.

After analyzing the optimal solutions of a large range of instances, we observed that most solutions are, in fact, 1-gfrp valid, i.e., no vertex not on ∂P had in-degree or out-degree larger than 1 in optimal solutions. Hence, we expect that most solutions are

ρ^∗_i

ρ^∗_j

i

j

ρ_ij ρ_ji

Figure 6.7: Clique constraints composition.

likely not to make use of crossing paths, as the ones shown in Figure 6.8. We can try to prevent crossing between two paths by joining them at their crossing point and reversing the direction of one of them, if necessary. Nonetheless, we could have an instance, like the one depicted in Figure 6.8, where it is not possible to build path φ₁ and then the reverse of path φ₃, as the fire will reach a point in the beginning of φ₃ before it can be successfully constructed. For a more concrete example, it is possible to tweak the fire and construction speeds for the instance depicted in Figure 6.6 (left) so that an optimal 1-gfrp solution is composed by paths ζ₁ = a, b, f and ζ₂ = d, c, e, while the global optimum is comprised of paths ζ₃ = a, b, c, e and ζ₄ = d, c, b, f .

In order to solve the 1-gfrp, we need to add Constraints (6.15) and (6.16) to our core gfrp model described earlier. Constraints (6.15) enforce the vertices’ maximum degree, preventing barrier paths from crossing each other or self-intersecting, like the optimal solution for the instance in Figure 6.6 (left). The right-hand side of these constraints can be replaced by k if we want to solve the general k-gfrp case. Constraints (6.16), although not required to solve the 1-gfrp, force that once we construct a barrier, we can only protect one of its incident faces, x_g or x_h. This prevents a barrier from being constructed inside the region guarded by an already constructed path, thus decreasing the set of feasible solutions. Notice that Constraints (6.16) are only valid for the 1-gfrp.

X

e∈Adj⁺(v)

y_e= X

e∈Adj⁻(v)

y_e ≤ 1 ∀v ∈ V^−→_B \ ∂P (6.15)

x_g + x_h+ y_i+ y_j ≤ 2 ∀(g, h) ∈ E_F | {i, j} ⊆ Adj^−→_B(g) ∩ Adj^−→_B(h) (6.16) The 1-gfrp admits an additional barrier discarding algorithm which is based on a primal bound β on the burned area. Consider the directed counterpart G^−→_F = (F, E^−→_F) of the graph G_F, defined in Section 6.2, to which we assign weights to each edge (f, g) ∈ E^−→_F to be the area of the face dual of f . The algorithm starts by computing the shortest paths over G^−→_F from the vertex corresponding to the face that holds the fire source to all other vertices. With these shortest paths, we can assign to a barrier i a minimum burned area β_i

φ₁

φ₂ φ₃

φ₄ r

Figure 6.8: gfrp valid solution that is not 1-gfrp conformant.

corresponding to the length of the minimum shortest path to a vertex whose corresponding face in A is incident to i. If β_i > β, it is certain that no optimal solution constructs i, as we already know a valid solution where the burned area is smaller than the minimum area burned in any solution that constructs i. The overall time complexity is defined by the shortest path algorithm which has complexity O(|E^−→_F| + |F | log |F |) ⊆ O(|B| log |B|) using a classical shortest path algorithm such as Dijkstra’s ([6]). For this algorithm to be applicable, each constructed barrier must be incident to the burned region in an optimum solution, as is the case of the 1-gfrp. To see that the algorithm is not suitable for the gfrp, consider the counter-example depicted in Figure 6.6 (left). Let ψf be the area of the region depicted in orange and ψ_z be the area of the trapezoid. Suppose we have an upper bound on the burned region, say β = ψ_f. Following the algorithm described above for barriers i ∈ M = {(c, e), (e, c), (b, f ), (f, b)} we would have βi = ψf + ψz > β.

The algorithm would then remove all barriers in M leaving no constructible barrier path, consequently leading to an optimum of 0. Notice, though, that the upper bound β = ψ_f, on the area of the burned region, also provides a positive lower bound on the protected area. This contradicts the optimality of the empty solution and, therefore, invalidates this discarding algorithm for the gfrp. Notwithstanding these facts, we will see in Section 6.7 that, in practice, optimal solutions for the gfrp are often optimal for the 1-gfrp.