2* ≥ z(Î*) + Σ λi (d; (γ) – 2) iЄN ΕΝ λί 2 1 1 2 1 2 1 3 5 100 5 4 1
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Consider the network in the following figure. Edges that are not pictured have a length of ∞.
Image attatched
(a) What is the optimal TSP tour on this network, and what is its total length, z∗? What is the optimal 1-tree Tˆ∗ on this network (“rooted” at node 1), and what is its total length, z(Tˆ∗)?
(b) Find the best Held-Karp lower bound you can. That is, find weights to add to one or more nodes so that z∗ and the right-hand side of the following formula are as close as possible.
z∗ ≥ z(Tˆ∗) + SUM{i∈N} λi(di(Tˆ∗) − 2) Image attatched
Here Tˆ* is the optimal 1-tree under the revised distance matrix, z(Tˆ*) is its cost anddi(Tˆ*) is the degree of node i in 1-tree Tˆ*.
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