Chapter 4. Network Topology Design
4.6. Binary Integer Linear Programming Transformation Methodology
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Fig. 4.5 shows the flow chart of profit maximization. The input parameters are G(V,E), S, R, W, C, and N’(V’,E’). N’(V’,E’)’ is set as the initial solution. H records the nodes that are probed and is set as null, initially. The output parameter is the network topology, G’(V’,E’). The purpose of total profit maximization is to maximize the total profit of G’(V’,E’) by adding the node which has the highest profit to G’, iteratively.
Let vi represents the neighbor node of G’ that has the highest profit and has not been included into H. In Step1, the process selects vi to include into H and G’.
In Step 2, if G’ satisfies all constraints after including vi into G’, then goes back to Step 1 when |V’| < C. Otherwise, G’ is not a valid solution, then removing vi from G’ and goes back to Step 1.
If all neighbor nodes of G’ are included into H and |V’| < C, it means that the process is caught into a local optimal. In order to obtained better solutions by trying another probing sequence, the process randomly removes some nodes from G’ and H and goes back to Step 1.
The process will be repeated until |V’| = C. G’(V’,E’) is the solution of TDHA. Pseudo code of total profit maximization process is listed in Appendix I.
4.6. Binary Integer Linear Programming Transformation Methodology
To evaluate our heuristic algorithm, we use Binary Integer Linear Programming (BILP) to obtain optimal solutions, although optimal algorithms are not practical in real DRO. In order to translate CCN CNTD problem into BILP problem, some new variables are introduced.
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The objective function and the constraints listed in section 4.3 are rewritten as follows.
bvi is a binary variable, where i=1,2,…,n. bvi =1 represents that vi is selected;
bei,j is a binary variable, where i,j=1,2,…,n. bei,j =1 represents that ei,j is selected;
Adiv(i) denotes the set of links diverge from node vi ;
Aconv(i) denotes the set of links converge to node vi;
Sum(Adiv(i))=∑𝑗=0,1,2,…,𝑛𝑏𝑒𝑖,𝑗 denotes the number of edges diverge from node vi;
Sum(Aconv(i))=∑𝑗=0,1,2,…,𝑛𝑏𝑒𝑗,𝑖 denotes the number of edges converge to node vi;
DG(vi) = Sum(Adiv(i))+ Sum(Aconv(i)) denotes the degree of vi;
Pi,j = {pzi,j| z =1,2,3,…,| Pi,j |} denotes the set of paths from vi to vj and pzi,j represents the z-th path from node i to j;
bPi,j = {bpzi,j| z =1,2,3,…,| Pi,j |} denotes the set of binary variables of the paths from vi to vj and bpzi,j represents the binary variable of path pzi,j;
(1) Transformation of the relationship between nodes and edges
Because the network topology G’ is a connected graph, the nodes and edges have the following relationship. If vi is not selected, then neither a converge nor a diverge edge of vi
can be selected. If any converge edge of vi is selected, vi must be selected.
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Binary linear equations that express the relationship between nodes and edges are listed as follows.
bei,j – bvi ≤ 0 ,for all vi ϵ V; ei,j ϵ Adiv(i) --- (4.7) bej,i – bvi ≤ 0 ,for all vi ϵ V; ej,i ϵ Aconv(i) --- (4.8) vi –Sum(Aconv(i)) ≤ 0 ,for all vi ϵ V --- (4.9)
Figure 4.6. Example of relationship between nodes and edges
Fig. 4.6 shows an example of the relationship between nodes and edges, in which, Aconv(i)
={ea,i, eb,i} and Adiv(i) ={ei,c, ei,d}. If vi is not selected (bvi=0), then ea,i, eb,i, ei,c and ei,d must not be selected (bea,i= beb,i= bei,c = bei,d=0). On the other hand, if vi is selected (bvi=1), then ea,i, eb,i, ei,c and ei,d can be either selected or not selected (0 ≤ bea,i, beb,i, bei,c , bei,d ≤ 1). These relationship are expressed in formula 4.7 and 4.8.
Furthermore, if vi is not selected (bvi=0), then all converge edges of vi cannot be selected (Aconv(i)=0). If vi is selected (bvi=1), then at least one converge edge of vi must be selected (bvi
=1, Aconv(i)= bea,i+beb,i bvi ≤ Aconv(i)). This relationship is expressed in formula 4.9.
(2) Transformation the relationship between edges and paths
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Paths are composed by edges. Relationship between a path and its edges are expressed in formula 4.11 and 4.12. Formula 4.11 shows that if a path, pzx,y, is not selected (bpzx,y=0), then there exist an edge that belongs to path pzx,y is not be selected. Formula 4.12 shows that if a path, pzx,y, is selected (bpzx,y=1), then all edges that belongs to path pzx,y must be selected.
Sum(pzx,y)- bpzx,y ≤ | pzx,y |-1 ,for all vx ,vy ϵ V; pzx,y ϵ Px,y --- (4.10) bpzx,y – bei,j ≤ 0 ,for all vx ,vy ϵ V; pzx,y ϵ Px,y;
ei,j ϵ pzx,y
--- (4.11)
Figure 4.7. Example of relationship between paths and edges
Fig. 4.7 shows an example of the relationship between paths and edges, in which, p1x,y={ex,d, ed,e, ee,y}, p2x,y={ex,c, ec,y}, p3x,y={ex,a, ea,b, eb,y}, Sum(p3x,y) = bex,a + bea,b + beb,y and | p3x,y |=3.
Referring to formula 4.10, if path, p3x,y, is not selected (bp3x,y=0), then Sum(p3x,y)- bp3x,y ≤ | p3x,y |-1. It implies that bex,a + bea,b + beb,y ≤ 2. The equation means that at least one edge belonging to p3x,y cannot be selected.
Referring to formula 4.11, if path, p3x,y, is selected (bp3x,y=1), then bp3x,y – bei,j ≤ 0, for all ei,j
belongs to p3x,y. It implies 1 ≤ bei,j . All edges belong to p3x,y must be selected.
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(3) Transformation of hop count limit
Let Px,y(h) = {pzx,y| pzx,y ϵ Px,y and | pzx,y |≤h}. Px,y(h) is the set of paths from vx to vy whose length is less than or equal to h. When a pivot nodes vy is selected, then there exist a paths from root the vy, pzx,y, whose length is less than or equal to hop count limit, U. The hop count limit is showed in formula 4.12.
bvy - Sum(Proot,y(U)) ≤ 0 , for all vy ϵ Φ; pzroot,yϵ Proot,y;
|pzroot,y| ≤ U
--- (4.12)
(4) Transformation of resource constraints
The constraint that the number of selected nodes is equal or less than the number of CRPs and this constraint is expressed in formula 4.13. Formula 4.14 represents that the degree of vi is equal or less than the number of antennas, D, in each CRP. The degree of vi is equal to Sum(Aconv(i))+ Sum(Adiv(i)). Because ei,j and ej,i denotes the same wireless link, ei,j and ej,i
cannot be selected simultaneously to avoid double count of wireless links (refers to formula 4.15).
Sum(V) = C --- (4.13)
Sum(Aconv(i))+ Sum(Adiv(i)) ≤ D ,for all vi ϵ V --- (4.14) bei,j + bej,i ≤ 1 ,for all vi, vj ϵ V --- (4.15)
(5) Transformation of minimum disjoint path constraints
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Each pivot node requires a minimum of Q disjoint paths to meet the reliability constraint.
Proot,y(ei,j) denotes the set of paths from root to vy that have common edge ei,j. Formula 4.16 represents that at most one of paths belonging to Proot,y(ei,j) can be selected due to the disjoint path constraint. Formula 4.17 represents that a minimum of Q paths from root to vy must be selected.
Sum(Proot,y(ei,j)) ≤ 1 for all vy ϵΦ; ei,j ϵ E;
| Proot,y(ei,j)|>1
--- (4.16)
Q - Sum(Proot,y) ≤ 0 for all vy ϵΦ --- (4.17)
Figure 4.8. Example of multiple disjoint paths
Referring to Fig. 4.8, p1x,y={ex,d, ed,c, ec,y}, p2x,y={ex,c, ec,y} and p3x,y={ex,a, ea,b, eb,y}. p1x,y and p2x,y have common edge ec,y, Px,y(ec,y) = {p1x,y, p2x,y}. Therefore, they cannot be selected simultaneously. If the number of outgoing paths, Q, is 2, then the feasible solutions are either {p1x,y, p3x,y} or {p2x,y, p3x,y}. If Q = 3, then no feasible solution can be found.
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(6) Transformation of path reliability constraints
If the path reliabilities of pivot nodes are given, then the number of disjoint paths can be computed. Let the path reliabilities of pivot nodes be greater than 1 -α. Let Rel(X) denotes the reliability of X and Fail(X) denotes the failure probability of X where Fail(X)=1-Rel(X).
The path reliability from root to vy, Rel(pzroot,y), is the probability that all edges belonging to pzroot,y are functional. Rel(pzroot,y) is equal to the multiplication of the reliabilities of the edges that belong to pzroot,y.
Rel(pzroot,y) = ∏ Rel(ei,j), for all ei,j ϵ pzroot,y
Fail(pzroot,y) = 1- Rel(pzroot,y)
Failure probability of pivot vy, Fail(vy), is equal to the multiplication of the failure probability of the paths that belong to Proot,y∩G’.
Fail(vy) = ∏ Fail(pzroot,y), for all pzroot,y ϵ Proot,y∩G’
Transferring the equation of Fail(vy) to a binary linear equation by adding logarithm and binary variables of paths, bpzroot,y , for all pzroot,y belong to Proot,y.
log Fail(vy) = log ∏ Fail(pzroot,y) , for all pzroot,y ϵ Proot,y∩G’
log Fail(vy) =Σ log Fail(pzroot,y) , for all pzroot,y ϵ Proot,y∩G’
log Fail(vy) =Σ[log Fail(pzroot,y)]×bpzroot,y , for all pzroot,y ϵ Proot,y
Since Fail(vy) is less than or equal toα.
Σ[log Fail(pzroot,y)]×bpzroot,y ≤ logα, for all pzroot,y ϵ Proot,y
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Referring to formula 4.18, the binary linear equation of computed number of disjoint paths is showed.
Σ[log Fail(pzroot,y)]×bpzroot ≤ logα for all vy ϵΦ;
and pzroot,y ϵ Proot,y
--- (4.18)
Take Fig. 4.7 as an example, assuming the reliability of each edge is 0.9, then the failure probabilities of paths, p1root,y, p2root,y and p3root,y can be computed as follow.
Fail(p1root,y) = 1- Rel(p1root,y) = 1 – Rel(ex,d)×Rel(ed,c)×Rel(ec,y) = 1-0.93=0.27;
Fail(p2root,y) = 1- Rel(p2root,y) = 1 – Rel(ex,c)×Rel(ec,y) = 1-0.92=0.19;
Fail(p3root,y) = 1- Rel(p3root,y) = 1 – Rel(ex,a)×Rel(ea,b)×Rel(eb,y) = 1-0.93= 0.27;
The relaibility of pivot node, vy, is computed as follow.
log Fail (vy) = Σ[log Fail(pzroot,y)]×bpzroot , z=1,2,3
log Fail (vy)=bp1root,y×log Fail(p1root,y) + bp2root,y×log Fail(p2root,y) + bp3root,y×log Fail(p3root,y)
log Fail (vy)= bp1root,y×log 0.27 + bp2root,y×log 0.19 + bp3root,y×log 0.27
log Fail (vy) = (-0.569) bp1root,y + (-0.721) bp2root,y + (-0.569) bp3root,y When applying path {bp1root,y, bp3root,y } as outgoing paths, then Rel (vy)=92.71%
log Fail (vy) = -1.137
Fail (vy) = 7.29%
Rel (vy) =1- Fail (vy) = 92.71%
When applying path {bp2root,y, bp3root,y } as outgoing paths, then Rel (vy)=94.87%
log Fail (vy) = -1.29
Fail (vy) = 5.13%
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Rel (vy) =1- Fail (vy) = 94.87%
(7) Transformation of cross network constraints
Let coverx = {vi,vj,vk,…} denotes the set of BSs that cover the same area x with corresponding selection sequence 1,2,3,…. The order of vi in the selection sequence precedes that of vj. Formula 4.19 indicates that vi must be selected before vj.
bvj – bvi ≤ 0 for all vi, vj ϵ coverx; mi<mj --- (4.19)
(8) Transformation of topology type constraints
If G’ is a tree-type topology then, |V’|-1 = |E’|, since |V’|=Sum(V) and |E’|= Sum(E).
Referring to formula 4.20, the tree type topology constraint is that the number of node of G’
minus one must be equal to the number of edges of G’. If G’ is a mesh network, then no constraint is necessary.
Sum(V)- Sum(E) = 1 --- (4.20)
Let x denotes the solution of binary linear programming. x is a binary vector.
x = {…,bvi,…,bei,j,…,bpzroot,y,… }, for all vi ϵ V, ei,j ϵ E, vy ϵΦ and pzroot,y ϵProot,y
The objective function of BILP is expressed as follows.
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Maximize
f (x) = ∑ bvi × ri × g(mi), for all vi ϵ V Subject to
bei,j – bvi ≤ 0 , for all vi ϵ V; ei,j ϵ Adiv(i) --- (4.21) bej,i – bvi ≤ 0 , for all vi ϵ V; ej,i ϵ Aconv(i) --- (4.22) vi –Sum(Aconv(i)) ≤ 0 , for all vi ϵ V --- (4.23) Sum(pzroot,y)- bpzroot,y ≤ | pzroot,y |-1 , for all vy ϵ Φ; pzroot,y ϵ Proot,y --- (4.24) bpzroot,y – bei,j ≤ 0 , for all vy ϵ Φ; pzroot,y ϵ Proot,y;
ei,j ϵ pzroot,y
--- (4.25)
bvy - Sum(Proot,y(U)) ≤ 0 , for all vy ϵ Φ; pzroot,yϵ Proot,y;
|pzroot,y| ≤ U
--- (4.26)
Sum(V) = C --- (4.27)
Sum(Aconv(i))+ Sum(Adiv(i)) ≤ D , for all vi ϵ V --- (4.28) bei,j + bej,i ≤ 1 , for all vi, vj ϵ V --- (4.29) Sum(Proot,y(ei,j)) ≤ 1 , for all vy ϵΦ; ei,j ϵ E;
| Proot,y(ei,j)|>1
--- (4.30)
Q - Sum(Proot,y) ≤ 0 , for all vy ϵΦ --- (4.31) bvj – bvi ≤ 0 , for all vi, vj ϵ coverx; mi<mj --- (4.32)
Sum(V)- Sum(E) = 1 , If G’ is tree-type --- (4.33)
f(x) denotes the profit function. Constraints 4.21, 4.22 and 4.23 represent the relationship between nodes and edges. Constraints 4.24 and 4.25 represent the relationship between the outgoing paths of pivot nodes and edges. Constraint 4.26 excludes those paths violating the
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hop count limit constraints. Constraints 4.27, 4.28 and 4.29 represent the limits of the number of CRPs and the number of antennas in each CRP. Constraints 4.30 and 4.31 indicate that the pivots nodes should have Q disjoint outgoing paths. Constraint 4.32 indicates the cross network constraint. Constraint 4.33 becomes effective when G’ is a tree-type topology.
In this section, we propose a transformation methodology that transforms the CCN CNTD problem to a BILP problem which can be solved using any binary linear programming algorithm. This transformation methodology is not only applicable to CCN network topology design, but also applicable to other optimization problems. One major benefit of this methodology is that there exists some commercial software such as MatLab such that even non-professional programmers can easily solve the problems. Pseudo code of binary integer linear programming algorithm is listed in Appendix II.