Consider the Travelling Salesman Problem:
Given N cities connected by edges of varying weights. Given a city A what is the shortest path for visiting all the cities exactly once that returns back to A
and the Max Travelling Salesman Problem:
Given N cities connected by edges of varying weights. Given a city A what is the largest path for visiting all the cities exactly once that returns back to A
Notice the following: If we take all the edge lengths of regular Travelling Salesman and we compute their reciprocals for example if we have edge lengths: 1, 2, 3 we form a new problem with corresponding edges 1, 1/2, 1/3 we can then solve regular Travelling Salesman with Max Travelling salesman.
Does this work?
An equivalent notion is the following:
Given a set of numbers
$Q = {x_1, x_2, x_3 ... x_n}$
if a k element minimum of the set is
${y_1, y_2, y_3 ... y_k}$
such that $y_i$ are all members of Q
Then:
Given a set of numbers
$Q' = {\frac{1}{x_1},\frac{1}{x_2},\frac{1}{x_3}... \frac{1}{x_n}}$
The k element maximum of the set is:
${\frac{1}{y_1},\frac{1}{y_2},\frac{1}{y_3}... \frac{1}{y_n}}$
^how do you prove that?