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To show that it provides a 2-approximation solution:  First we know that the sum of all the edges weights replace_contentgt;=$ >=  optimal value.S ince value. Since  the cost is derived from the number of edges, it can’t exceed the sum of the edge weights. Also we know that if we reached a locally optimal solution, that means if you swap two vertices, the total cost must decrease. Let's call our two sets be call $A$ and $B$. If we are a locally optimum solution and we move the node u from A to B, we get:  \sum\nolimits_{P_i \in Paths(I)} Probes(P_{i})