I would like to know if there exists any good methods that can determine the following class of problems:
Suppose there exists $n$ given positive integers $y_1, y_2, \dots, y_n$ and positive integers $F, H$. Determine if there exists integers $x_1, x_2, \dots, x_n$ such that
$$x_1y_1 + x_2y_2 + \dots + x_3y_3 = F$$
under the constraint that
$$\mid x_1 \mid + \mid x_2 | + \dots + \mid x_n \mid = H$$
Indeed one can very if a solution exists or not using the Generalized Bezout's Identity. However this includes solutions that do not satisfy the above constraint.
I notice that one can represent the problem as asking whether or not there exists integer vectors $\vec{x}$ such that $\vec{x} \cdot \vec{y} = F$, under the constraint that the L1-Norm of $\vec{x}$ is equals to $H$, though I am not sure if this would be of much use.
I am currently using a brute-force algorithm that permutes the values of $x_i$ for given $H$ to determine the existence of solutions. Without doubt, such a method has unacceptably horrible performance for big values of $H$ and $n$.