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Assume a weighted graph G and a positive value k are given.

What is the complexity of finding a cycle with total weight k when G is Hamiltonian or Hamiltonian-connected?

pointing to papers and books is also welcome!
I wish it wouldn’t look as a homework!

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If we are asking for a simple cycle the problem is NP-complete by a reduction from the Hamiltonian cycle problem.

We want to find a Hamiltonian cycle in a graph $G$. We assign weight 1 to all edges of $G$, and add to this graph all the other edges with weight $\infty$. We have thus created a clique, which is obviously Hamiltonian, and we ask whether there exist a cycle with weight $n$ in it.

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  • $\begingroup$ Good idea, but I think your answer should change a little bit, because in this case someone can just traverse all edges in $O(n^2)$ (in a graph with $n$ vertices) and pick those edges with weight 1! I think it is necessary to build a Hamiltonian cycle with random total weight, and then add other edges with other random weights. Thanks... $\endgroup$ – marjoonjan Dec 12 '10 at 17:22
  • $\begingroup$ Of course, you're right. I meant $\infty$ instead of 0 -- I corrected my post. $\endgroup$ – Karolina Sołtys Dec 12 '10 at 17:26
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    $\begingroup$ $n$ works just as well as $\infty$ here. $\endgroup$ – Peter Shor Dec 12 '10 at 17:54

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