linear programming – When LP solution is ILP solution?

For many discrete problems, it’s natural to consider their continuous relaxations. A common case is when instead of $x_i in {0, 1}$ we allow $x_i in (0, 1)$. In certain cases, the original problem is an integer linear programming (ILP) problem, and its relaxed version becomes a linear programming (LP) problem, which we can efficiently solve.

Questions: Are there common techniques which show that, for a particular problem:

  1. An LP solution will always be an ILP solution?
  2. There exists an LP solution that is also an ILP solution?
  3. A particular LP algorithm (e.g. simplex method) finds an ILP solution.

By “solution”, I, of course, mean a vector on which the objective reaches its optimum.