p653.pdf - 13.6. Exercises 641 C-13.14 Consider the general...

13.6. Exercises 641
C-13.14 Consider the general optimization version of the TSP problem, where the un-
derlying graph need not satisfy the triangle inequality. Show that, for any fixed
value δ ≥ 1, there is no polynomial-time δ -approximation algorithm for the gen-
eral TSP problem unless P = NP .
Hint: Reduce H
AMILTONIAN
-C
YCLE
to this problem by defining a cost function
for a complete graph H for the n -vertex input graph G so that edges of H also in
G have cost 1 but edges of H not in G have cost δ n more than 1.
C-13.15 Consider the special case of TSP where the vertices correspond to points in the
plane, with the cost defined on an edge for every pair ( p , q ) being the usual
Euclidean distance between p and q . Show that an optimal tour will not have any
pair of crossing edges.
C-13.16