FORMULATIONS AND HEURISTICS FOR MINIMAL SPANNING TREES WITH MINIMUM DEGREE CONSTRAINTS Pedro Martins∗ , Mauricio C. de Souza§ , and Ana M. de Almeida‡ ∗ Instituto
Polit´ecnico de Coimbra Coimbra, Portugal e-mail:
[email protected]
§ Universidade
Federal de Minas Gerais Belo Horizonte, Brasil e-mail:
[email protected] ‡ Universidade
de Coimbra Coimbra, Portugal e-mail:
[email protected]
1
INTRODUCTION
Let G = (V, E) be a connected undirected graph, where V = {1, . . . , n} is the set of nodes and E the set of edges. Consider that there are associated positive costs, ce , to each edge e ∈ E. Given a positive integer valued function d : V → N on the nodes, the Min-Degree Constrained Minimum Spanning Tree (md-MST) problem consists in finding a spanning tree T of G with minimum total P edge cost, given by e∈T ce , and where each node i ∈ V either has degree at least d(i) or is a leaf node. Practical applications to the md-MST problem may occur in cases where one needs to identify a set of places (nodes) that centralize incidence with other entities (peripheral nodes), in a way that a place (node) can only assume a central status if assigned to, at least, d other places (nodes). Otherwise it must be a peripheral entity (leaf node). Therefore, the central nodes can characterize central distribution places, or centralized communication devices, while the peripheral nodes act as individual consumers or clients. This problem was introduced in Almeida, Martins and Souza [1] and showed the be N P -hard for d ≥ 4. Considering d ≥ 4, we can reduce the k–Dimensional Matching Problem, where k ≥ 3, to the md-MST, using a constructive approach analogous to the proof of Partition into Triangles found in [2]. Complexity for d = 3 is still open, while for d ≤ 2 the problem becomes the MST. 2
FORMULATIONS TO THE md-MST PROBLEM
We propose two extended formulations to the md-MST problem, using flow based models. These formulations are defined on a directed version of graph G, obtained by replacing each edge {i, j} by two directed arcs (i, j) and (j, i) both having the same cost as the original edge. We define a special node r taken from V , from which only outward arcs will be kept in the graph. We denote the direct graph by Gr = (V, Ar ), where Ar = {(i, j) : i ∈ V and j ∈ V \ {i, r}}. Node r acts as a root in any feasible arborescence of Gr , generating the whole flow sent into the network. We model the md-MST problem with binary variables xij for all (i, j) ∈ Ar , where xij = 1 if arc (i, j) is in the solution, and 0 otherwise. We also define the set of node-variables ki , where ki = 1 if i is a central-node, and ki = 0 when i is a leaf-node.
Considering Xr as the set of incidence vectors that characterize spanning trees of Gr , see Magnanti and Wolsey [3], the formulation md-F is written as follows: X cij xij (1) minimize (i,j)∈Ar
(d − 1)kr ≤
X
xrj − 1 ≤ (n − 2)kr
(2)
j∈V \{r}
(d − 1)ki ≤
X
xij ≤ (n − 2)ki
∀i ∈ V \ {r}
(3)
j∈V \{i,r}
x ∈ Xr ⊂ {0, 1}(n−1) ki ∈ {0, 1}
2
(4) ∀i ∈ V
(5)
Constraints (2) and (3) define lower and upper bounds on the number of outward arcs incident in each node i. Node r is a leaf when there is a single arc diverging from it, while any other leaf-node has a single inward arc. The general model md-F can be used to produce various formulations by using known characterizations of spanning trees, see [3]. In the present work, we consider only single-commodity and multicommodity flow models in order to characterize set Xr . We define the following two other sets of valid inequalities for the md-F model: xij ≤ ki , i, j ∈ V \{r}, i 6= j
(6)
establishing that if there is an outward arc incident in node i, then it must be a central-node, while if i is a leaf-node, then there cannot be any arc leaving node i; and X n−2 ki ≤ (7) d−1 i∈V
defining an upper bound to the number of central-nodes in any feasible solution to the md-MST problem. 3
HEURISTICS FOR THE md-MST PROBLEM
We first propose a greedy constructive heuristic for the md-MST problem. Then, with the constructive heuristic on hand, we exploit an Enhanced Second Order (ESO) algorithm, proposed by Martins [4], within the improvement phase of the Variable Neighborhood Search (VNS) method, proposed by Mladenovi´c and Hansen [5], substituting the most common local search methodology. The constructive heuristic is an adaptation to the md-MST problem of Kruskal’s algorithm. To do this, we need to test, in each iteration, not only if the entering edge (i, j) forms a cycle, but also if it is still possible to reach a tree with all central nodes feasible lower degree, after including edge (i, j). An edge (i, j) will be allowed for inclusion if the number of nodes-degrees still remaining for completion, assuming the inclusion of (i, j), is sufficient to provide feasibility. The two following quantities are compared: • 2(n − k − 1): the number of nodes-degrees to still furnish the current forest, after including the k th edge;
• need: number of nodes-degrees that the current forest needs to reach a final md-MST feasible solution, after including the k th edge. Thus, an edge (i, j) is allowed to be included if it does not form a cycle in the current forest and if, assuming the k th edge inclusion, we have that need ≤ 2(n − k − 1). The need value, assuming the k th edge inclusion, leading to nodes partition V1 , V2 , . . . , V(n−k) , is defined by X X X need = (feasible components) + (d − dT (i)) + (d − 1) (8) i∈V : 1
