TSP(1,2), the Traveling Salesman Problem with distances 1 and 2, is
the problem of finding a tour of minimum length in a complete
weighted graph where each edge has length 1 or 2. Let $d_o$ satisfy
$0<d_o<1/2$. We show that TSP(1,2) has no PTAS on the set of
instances where the density of the subgraph spanned by the edges with
length 1 is bounded below by $d_o$. We also show that LONGEST PATH
has no PTAS on the set of instances with density bounded below by
$d_o$ for all $0 < d_o < 1/2$.