Consider we have $n$ vertices, $v_1,\ldots,v_n$. We have two positive values $(a_i,b_i)$ associated with each $v_i$. The edge weight $w(v_iv_j)=a_ia_j+b_ib_j$.
Is it NP-hard to solve the traveling salesman problem on this graph?
In the special case where all the $b_i$ are $0$, this is solvable in $O(n\log n)$ time. Because this is the traveling salesman problem on a symmetric product matrix. See section 5.2 in this survey.
