Clarification for binary search in solving optimal TSP when a polynomial algorithm with a budge exists Below is Question 8.1 in Algorithms by Dasgupta et al.

There's a solution to this problem that uses binary search from here. Pasting the answer for posterity.

My questions are:

When they say input size, are they just referring to edges or both
edges and vertices?
Is it correct to say that we make at most $log( |E| log K)$ calls?
Does this work if the edge weights aren't integers, like 10.4 etc?
Why is this NP-Hard as the solution mentions? I thought that Traveling Salesperson Problem was NP-Complete

This question on stackoverflow asks for clarification of binary search use but doesn't answer the above question.
Input size refers to memory size needed to represent the input. In this case, it means number of edges + number of vertices, and you could also add size of the weights (which is logarithmic in their values).
No. It is written that we do $\log T$ calls and $T \leq |E| K$ (and this value can be reached), so the number of calls is at most $\log (|E| K) = \log |E| + \log K$.
Yes. It is never used in the algorithm that weights must be integers.
A $NP$-complete problem is a $NP$-hard problem which also needs to be in $NP$. So when speaking about a $NP$-complete problem, it is not false to state that it is $NP$-hard. However, $NP$ is a set of decision problems, therefore, TSP-OPT being an optimization problem cannot be in $NP$. We can still speak about $NP$-hardness as it can be informally defined for a non-decision problem as &quot;a problem more difficult than all problems in $NP$&quot;. When you say &quot;I thought that Traveling Salesperson Problem was NP-Complete&quot;, it refers to the decision version, denoted TSP in your reference.