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

• Iskander Aliev
• Jesús A. De Loera
• Quentin Louveaux
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8494)

Abstract

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 n , 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.

Keywords

Polynomial Time Lattice Point Knapsack Problem Integer Solution Numeric Semigroup
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer International Publishing Switzerland 2014

Authors and Affiliations

• Iskander Aliev
• 1
• Jesús A. De Loera
• 2
• Quentin Louveaux
• 3
1. 1.Cardiff UniversityUK
2. 2.University of CaliforniaDavisUSA
3. 3.Université de LiègeBelgium

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