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Thank you for your answer @Ilya . In my problem, I actually need the vectors to have a minimum Hamming distance of $d$. Sorry, I did not mention this earlier in my problem. I have edited the problem now to include this concern. Can we tweak your response to accommodate this concern?
Minimum Spanning Tree is not an NP-Hard problem and Polynomial time algorithms like Prim and Kruskal exist for MST. What do you mean by approximation algorithm for a problem which is not NP-Hard?