Heuristics for Synthesizing Robust Networks with a Diameter Constraint

Algorithm 3

Synthesizing robust networks with maximum algebraic connectivity subject to diameter constraints.

Let 𝔉 be a set of cuts which must be satisfied by any feasible solution

(1) Input: A graph 𝐺=(𝑉,𝐸), 𝑤_𝑒, 𝑟, 𝐷, and a finite number of unit vectors, 𝑣_𝑖,𝑖=1,…,𝑀

(2) Choose any spanning tree satisfying the diameter constraint as an initial feasible solution, 𝑥^∗

(3)  𝜆←𝜆₂(𝐿(𝑥^∗))

(4)  loop

(5)  𝔉←∅

(6)   Replace the semi-definite constraint in (B.1) with 𝑣_𝑖^𝑇(𝐿(𝑥))𝑣_𝑖≥𝜆𝑣_𝑖^𝑇𝐼_𝑛̃𝑣_𝑖∀𝑖=1,…,𝑀, and with additional

   constraint, 𝑥 satisfying 𝔉 and solve the ILP.

(7)   if  the above ILP is infeasible  then

(8)    break loop  {𝑥^∗ is the optimal solution}

(9)   else

(10)   𝑥^∗ ← solution to the above ILP

(11)   𝛾^∗ ← 𝜆₂(𝐿(𝑥^∗))

(12)   if  𝐿(𝑥^∗)⪰^╱𝛾^∗(𝐼_𝑛̃)  then

(13)    Augment 𝔉 with a constraint (𝑣^∗𝑇𝐿(𝑥^∗)𝑣^∗)≥𝛾^∗(𝑣^∗𝑇𝐼_𝑛̃𝑣^∗) where 𝑣^∗ is the eigenvector corresponding to a negative

     eigenvalue of 𝐿(𝑥^∗)−𝛾^∗𝐼_𝑛̃.

(14)    Go to step 6.

(15)   end if

(16)  end if

(17)  𝜆←𝜆+𝜖{let 𝜖 be a small number}

(18) end loop