Checking containment using integer programming

Question

For this question, a region is a subset of Z^d defined by finitely many linear inequalities with integer coefficients, where Z^d is the set of d-tuples of integers. For example, the set of pairs (x, y) of non-negative integers with 2x+3y >= 10 constitutes a region with d=2 (non-negativity just imposes the additional inequalities x>=0 and y>=0).

Question: is there a good way, using integer programming (or something else?), to check if one region is contained in a union of finitely many other regions?

I know one way to check containment, which I describe below, but I'm hoping someone may be able to offer some improvements, as it's not too efficient.

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Here's the way I know to check containment. First, integer programming libraries can directly check if a region is empty: in integer programming terminology (as I understand it), emptiness of a region corresponds to infeasibility of a model. I have coded up something using the gurobi library to check emptiness, and it seems to work well in practice for the kind of regions I care about.

Suppose now that we want to check if a region X is contained in another region Y (a special case of the question). Let Z be the intersection of X with the complement of Y. Then X is contained in Y if and only if Z is empty. Now, Z itself is not a region in my sense of the word, but it is a union of regions Z_1, ..., Z_n, where n is the number of inequalities used to define Y. We can check if Z is empty by checking that each of Z_1, ..., Z_n is empty, and we can do this as described above.

The general case can be handled in exactly the same way: if Y is a finite union of regions Y_1, ..., Y_k then Z is still a finite union of regions Z_1, ..., Z_n, and so we just check that each Z_i is empty. If Y_i is defined by m_i inequalities then n = m_1 * m_2 * ... * m_k.

So to summarize, we can reduce the containment problem to the emptiness problem, which the library can solve directly. The issue is that we may have to solve a very large number of emptiness problems to solve containment (e.g., if each Y_i is defined by only two inequalities then n = 2^k grows exponentially with k), and so this may take a lot of time.

Answer 1

You can't really expect a simple answer. Suppose that A is defined by all constraints of the form 0 <= x_i <= 1. A can be thought of as the collection of all possible rows of a truth table. Given any logical expression of the form e.g. x or (not y) or z, you can express it as a linear inequality such as x + (1-y) + z >= 1 (along with the 0-1 constraints). Using this approach, any Boolean formula in conjunctive normal form (CNF) can be expressed as a region in Z^n. If A is defined as above and B_1, B_2, ...., B_k is a list of regions corresponding to CNFs then A is contained in the union of the B_i if and only if the disjunction of those CNFs is a tautology. But tautology-checking is a canonical example of an NP-complete problem.

None of this is to say that it can't be usefully reduced to ILP (which itself is NP-complete). I don't see of any direct way to do so, though I suspect that some of the techniques used to identify redundant constraints would be relevant.