Suppose, we have a finite set of generators $G \subset \mathbb{Z}^2_+$, where $|G|=n$.
And the set of interest is $P = \{ \sum_{i \in S} i | S \subseteq G\}$. The line of interest is $y=x$, using set notation $Q = \{(x, x)| x \in \mathbb{R}\}$.
Is there any efficient method to find $T = P\cap Q$?
Some attempts try to reduce the complexity to dimension 1.
by finding the norm vectors of $Q$: $n=(1, -1)$, we know $\langle n, q \rangle = 0, \forall q \in Q$.
Since $T \subset Q$, we also have $\langle n, t \rangle = 0, \forall t \in T $.
We can define a one dimensional set $H = \{ \langle n, g \rangle| g \in G \}$.
Now we only need to determine if $\{ \sum_{i\in S} i|S \subseteq H \}$ has zero element.