Constructive Heuristics for Min-Power Bounded-Hops Symmetric Connectivity Problem

Abstract

We consider a problem of constructing an energy-efficient bounded diameter communication spanning tree when the vertices are located on a plane, and the energy required to transmit a message between a pair of vertices is proportional to the squared distance between them. For this NP-hard problem, we have developed several approximate heuristic algorithms. The results of a posteriori analysis of solutions constructed by the proposed algorithms are presented.

The research of R. Plotnikov is supported by the Russian Science Foundation (project 18-71-00084, Sections 3, 4, and 5). The research of A. Erzin is supported by the Russian Science Foundation (project 19-71-10012, Sections 1 and 2).

Appendices

Appendix A: Algorithm Descriptions

Appendix B: MILP Formulation

We propose a MILP formulation of the BHMPSCP that is generally based on the ILP formulation of the BDMST from [23]. Let us construct such directed graph $G^{\prime } = (V, A)$ from the given graph $G = (V, E)$, that each edge $(i,j) \in E$ corresponds to two oppositely directed arcs $(i, j) \in A$ and $(j,i) \in A$ with the same weight $c_{ij} = c_{ji}$. Let us introduce the following variables:

• $u_{il} = 1$ if the depth of vertex i in a tree equals l and $u_{il} = 0$ otherwise, $i = 1, ..., n; l = 0, ..., \lfloor D/2 \rfloor$;

• $x_{ij} = 1$ if the arc $(ij) \in A$ belongs to the tree and $x_{ij} = 0$ otherwise, $i,j = 1, ..., n$;

• $C_i \ge 0$—maximum weight of an arc adjacent to the vertex i in the tree, $i = 1, ..., n$;

• $r_{ij} = 1$ if $(i,j) \in E$, and the vertices i and j are the centers in the solution to the problem, $i,j = 1, ..., n$. These variables are used only in the case when D is odd.

We propose two different formulations depending on the parity of D. If D is even then the problem is formulated in a following way:

$$\begin{aligned} \sum _{i = 1}^n{C_i} \rightarrow \min \end{aligned}$$

(1)

$$\begin{aligned} x_{ij}c_{ij} \le C_i, i,j = 1, ..., n \end{aligned}$$

(2)

$$\begin{aligned} x_{ij}c_{ij} \le C_j, i,j = 1, ..., n \end{aligned}$$

(3)

$$\begin{aligned} \sum _{l = 1}^{\lfloor D / 2 \rfloor }{u_{il}} = 1, i = 1, ..., n \end{aligned}$$

(4)

$$\begin{aligned} \sum _{i:(i,j) \in A}{x_{ij}} = 1 - u_{j0}, j = 1, ..., n \end{aligned}$$

(5)

$$\begin{aligned} x_{ij} \le 1 - u_{jl} + u_{i,l-1}, i,j = 1,...n: (i,j) \in A, l = 1, ..., \lfloor D / 2\rfloor \end{aligned}$$

(6)

$$\begin{aligned} \sum _{i = 1}^n{u_{i0}} = 1 \end{aligned}$$

(7)

In this formulation, constraints (2)–(3) bound the maximum weight of the arc in a tree for each vertex i by the corresponding variable $C_i$ which is used in the minimized function (1). Equation (4) guarantee that for each vertex the only value of depth is assigned. Equations (5) imply that each vertex, except the center, has the only incoming arc and only the center has the depth assigned to 0. Inequalities (6) reflect the fact that for each in-tree arc (i, j), the depth of i is less by 1 than the depth of j. Equation (7) ensures that the tree has the only center.

Formulation for the case when D consists of the same minimized function (1) and constraints (2)–(6). But it does not contain the equality (7) since in this case solution should have two centers. In addition, the following constraints are included to the problem formulation in the case when D is odd:

$$\begin{aligned} r_{ij}c_{ij} \le C_i, i,j = 1, ..., n \end{aligned}$$

(8)

$$\begin{aligned} r_{ij}c_{ij} \le C_j, i,j = 1, ..., n \end{aligned}$$

(9)

$$\begin{aligned} \sum _{j:(i,j) \in E}{r_{ij}} = u_{i0}, j = 1, ..., n \end{aligned}$$

(10)

$$\begin{aligned} \sum _{i = 1}^n{u_{i0}} = 2 \end{aligned}$$

(11)

In this formulation, the inequalities (8)–(9) bound below the appropriate variables $C_i$ by the weight of the edge that connects two centers. Equations (10) imply that only the centers (i.e., the vertices with zero depth) are connected by the special edge that is defined by the variables $r_{ij}$. And, finally, the equality (11) guarantees that there are two centers in solution.

Appendix C: Tables with Experiment Results

Table 1. Comparison of the experiment’s results obtained by different heuristics with CPLEX.

Table 2. Comparison of the experiment’s results obtained by different heuristics.