Chapter 4 BRANCH-AND-CUT METHOD FOR HVRPUD
4.3 Separation algorithm for comb inequalities
Several methods have been proposed to identify violated comb inequalities on the support graph for the TSP and these works have been later adapted to the VRP (see [3],[12],[40],[51],[54] and [62]). As mentioned in Chapter 2, the blossom valid inequality is a special case of the comb inequality where each tooth can contain only two vertices (one in common with the handle and one outside of the handle). There are also a few algorithms for separating this class of valid inequalities (see [3] and [50]). Recall the comb valid inequalities for a LP solution vectorx⇤:
x⇤(E(H)) + s X t=1 x⇤(E(Tt))|H|+ s X t=1 (|Tt| 1) d 1 2se. (4.7) as mentioned in Chapter 2, the comb inequalities can be stated in the below form:
x⇤( (H)) + s
X
t=1
x⇤( (Tt)) 3|T|+ 1 (4.8)
We first briefly explain two existing algorithms and then a new separation algorithm will be proposed.
Connected component method This method identifies bi-connected compo- nents of the✏-support graph. A✏-support graph is obtained by deleting edges with weights less than ✏ or greater than 1 ✏. Let us start this heuristics with two definitions we need later.
Definition 5. k-connected graph: A graph is k-connected if k vertices (along with their adjacent arcs) must be removed to disconnect the graph.
Two equivalent definitions of the above definition for the bi-connected graph are as follows. A connected graph is bi-connected if there are two paths between each two vertices or a connected graph is biconnected if the removal of any single vertex (and all edges incident on that vertex) cannot disconnect the graph.
Definition 6. Articulation points (cut vertex): any vertex whose removal (together with removal of any incident edges) results in a disconnected graph.
To identify candidate handle and teeth, the ✏-support graph is first con- structed, then each bi-connected component with at least three vertices is a candidate handle. The teeth are comprised of any vertices on the original support graph which are connected to the vertices in the bi-connected components. To find bi-connected components of the✏-support graph, the Depth First Search (DFS) algorithm can be applied (see [25] for the DFS algorithm).
Shrinking method The above procedure provides us configurations in the sup- port graph to set up the blossom or 2-matching inequalities. Combining algorithms developed for blossom inequalities with shrinking procedures can lead to e↵ective algorithms for general comb inequalities on the original graph. For instance, if there is a path P from vertex s to vertex t such thatxe = 1 fore 2E(P) (where E(P) is the set of edges on path P), then we can shrink path P and replace the whole path with an edge having weight xst = 1 ([3]). Several types of shrinking proce- dures have been suggested based on this simple idea by Padberg and Grotschel [60], Grotschel and Holland [40] and Padberg and Rinaldi [61]. Here we present Grotschel and Holland’s procedure as their procedure has been reported to be one of the most successful procedures and also easy in implementation ([3] and [50]).
Grotschel and Holland [40] propose five rules for shrinking as follows:
1. Given a path P of 1-edges (i.e. xe = 1 fore2E(P)) betweensandt, replace P with a single edge (s t) with xst= 1.
3. GivenS✓V so that|S|= 4 andx( (S)) = 2, then shrinkS to a single vertex. 4. Given {s, t, u, v} ✓ V so that xst = xvu = 1 and x({s, t}{u, v}) = 1, then
shrink {s, t}to {u, v}. See Figure 4.3.
5. Given {u, v, w} ✓ V so that xuv = 1 and x({s, t}{u, v}) 0.5, then shrink {u, v}. v u w 1 ↵ 1 ↵ w
Figure 4.2: 1-edge shrinking in a triangle
s t u v ↵ 1 ↵ ✓
Figure 4.3: 1-edge shrinking in a square
A greedy approach In this section, we propose a new greedy procedure to identify violated comb inequalities for the HVRP. The main idea of this procedure is to find out which configuration of a given set of vertices violates a valid inequality or is most
likely to violate a valid inequality. Similar to our proposed heuristic for the SEC, this heuristic is also motivated by shrinking procedure. In the proposed method, shrinking process is guided in the sense that even if it does not identify any violated inequality in an iteration, it forms a setting of a comb which is more likely to be violated in next steps. In standard shrinking procedures, as mentioned, we need to have 1-edges, whereas in our proposed method, there is no need to necessarily have 1-edges.
The heuristic starts with an initial comb. An initial comb configuration can be set up using the connected component method or algorithms developed for blossoms. LetCbe the set of vertices forming the comb. Then, we would like to find out moving a given vertex i 2 Vc\C to which sub-set (H \T, T \H or T \H) can lead to a violated valid inequality. If no violated constraint is found in this step, the vertex is added to a subset which is most likely to lead to a violated inequality in next steps. The procedure can be explained with a simple example.
Example Let assume the following initial comb configuration is given and let as- sume that the weight of each edge is one (see Figure 4.4):
H ={2,4,6}, T1={1,2}, T2={3,4}, T3={5,6}, Vc\C={7}.
We would like to find out moving vertex 7 to which sub-set leads to the highest reduction in the left-hand side(lhs) of (4.8). The smaller the lhs is, the more likely it
T1\H T1\H H \T 1 2 3 4 5 6 7
Figure 4.4: A comb consists of a handle and three teeth
is to lead to a violated inequality. Vertex 7 can be moved to either of following sets:
H \T, T1\H, T2\H, T3\H, T1\H, T2\H or T3\H
For instance, if the vertex is moved toT3\H, the value of the comb will be reduced by 2, because the edge (6-7) now intersects only one sub-set border and the edge (5-7) does not intersect any sub-set border any more.
4.4
Concluding remarks
We studied several separation procedures for the SEC and comb inequalities for the HVRPUD and proposed two greedy algorithms for these two types of inequalities. This study addresses the second research question. Indeed, we employ methods within mixed-integer programming to solve the HVRPUD.
rithm for the SEC is relatively efficient, the comb inequalities do not do anything good for our problem. In addition to the proposed algorithm, we implemented and tried the other separation algorithms in the literature but no improvement was achieved. This is in contrast with performance of the comb inequalities for the CVRP. But as the problem complexity increases, the performance of the comb valid inequalities significantly reduces. Belenguer et al. [13] report the similar performance for the comb inequalities.