Heuristics for Synthesizing Robust Networks with a Diameter Constraint

Algorithm 2

Tabu search heuristic.

Let 𝑠^∗ be the best optimal solution found by the algorithm and let 𝜆^∗₂ be the algebraic connectivity of 𝑠^∗. Let 𝑚 denote the

number of edges in the initial feasible solution (i.e., 𝑚∶=|𝑉|−1). Also, let 𝑀𝑎𝑥𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠 represent the maximum number

of iterations allowed in the tabu heuristic.

(1)  𝒯₀← Initial feasible solution satisfying diameter constraints

(2)  𝑠∶=𝒯₀, 𝜆^∗₂∶=𝜆₂(𝒯₀), 𝑠^∗∶=𝒯₀, 𝑇 is an empty tabu list which can store at most 𝑚 solutions.

(3)  for  𝑖=1 to 𝑀𝑎𝑥𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠  do

(4)  for  edge 𝑒_𝑘 in 𝑠  do

(5)    Construct the set of edges 𝑁(𝑠,𝑒_𝑘)

(6)   end for

(7)   for  any spanning tree 𝑠^′ such that 𝑠^′ contains exactly one edge from each of the sets in {𝑁(𝑠,𝑒₁),…,𝑁(𝑠,𝑒_𝑘)}  do

(8)    if  𝑠^′ is feasible  then 

(9)     if  𝜆₂(𝑠^′)>𝜆^∗₂  then

(10)     𝑠=𝑠^′

(11)     𝑇←𝑇⋃{𝑠^′}; 𝑇 only stores at most 𝑚 solutions from iterations 𝑖−𝑚+1,…,𝑖−1,𝑖

(12)     𝜆^∗₂=𝜆₂(𝑠), 𝑠^∗=𝑠

(13)     break

(14)    else

(15)     if  𝑠^′ is not in 𝑇  then

(16)      𝑠=𝑠^′

(17)      𝑇←𝑇⋃{𝑠^′}; 𝑇 only stores at most 𝑚 solutions from iterations 𝑖−𝑚+1,…,𝑖−1,𝑖

(18)      break

(19)     end if

(20)    end if

(21)   end if

(22)  end for

(23)  if  𝑠≠𝑠^′ then

(24)   break //The heuristic terminates here if there is no feasible tree 𝑠^′ in the neighborhood of 𝑠 or if every feasible 𝑠^′

    with a lower algebraic connectivity (< 𝜆^∗₂) is already in the tabu list.

(25)  end if

(26) end for

(27) output 𝜆^∗₂ and 𝑠^∗