Integer Programs with Prescribed Number of Solutions and a Weighted Version of Doignon-Bell-Scarf's Theorem

In this paper we study a generalization of the classical feasibility problem in integer linear programming, where an ILP needs to have a prescribed number of solutions to be considered solved. We first provide a generalization of the famous Doignon-Bell-Scarf theorem: Given an integer k, we prove that there exists a constant c(k, n), depending only on the dimension n and k, such that if a polyhedron {x : Ax ≤ b} contains exactly k integer solutions, then there exists a subset of the rows of cardinality no more than c(k, n), defining a polyhedron that contains exactly the same k integer solutions. The second contribution of the article presents a structure theory that characterizes precisely the set Sg≥k(A) of all vectors b such that the problem Ax = b, x ≥ 0, x ∈ Z, has at least k-solutions. We demonstrate that this set is finitely generated, a union of translated copies of a semigroup which can be computed explicitly via Hilbert bases computation. Similar results can be derived for those right-hand-side vectors that have exactly k solutions or fewer than k solutions. Finally we show that, when n, k are fixed natural numbers, one can compute in polynomial time an encoding of Sg≥k(A) as a generating function, using a short sum of rational functions. As a consequence, one can identify all right-hand-side vectors that have exactly k solutions (similarly for at least k or less than k solutions). Under the same assumptions we prove that the k-Frobenius number can be computed in polynomial time.