Name for subclass of ILP without any inequality constraints (including constraints on x)

Question

In "Myths and Counterexamples of Mathematical Programming" myth "IP Myth 21" says:

The problem of finding $x\in \mathbb{Z}$ such that $Ax=b$, where $A\in\mathbb{Z}^{m\times n}$ and $b\in \mathbb{Z}^m$, is NP-complete.

Which is incorrect (as all myths in this excellent resource are).

The key to this myth is that there are no bounds on $x$. (If we add $x≥0$, we have the standard form of a linear integer program, which is NP-complete.) This is the problem of solving linear diophantine equations for which Brown provides simple, polynomial-time algorithms

• Is there a name for this problem class in OR/Optimization?

• Is there a name for this problem class some linear equality constraints over $\mathbb{R}$ are added?

• Are you aware of any MILP solvers which find such subproblems and solve them using specialized routines?

• Are problems of the same form but with $\min ||x||$, $\min c^Tx$, $\min \sum_i (c_i x_i)^2 $, $\min x^T D x$ or $\min x^T D x + c^Tx$ where D positive definite as objective still solvable in polynomial time?

Answer 1

1. https://en.m.wikipedia.org/wiki/Diophantine_equation
2. any milp solver can handle it, but maybe something like mathematica/mable has special algortihms or one can use the algorithm described by the wiki entry

4a) https://en.m.wikipedia.org/wiki/Lattice_problem 4b) is i think polytime, either the problem is unbounded, infeasible or all solutions are optimal.