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Sparsity and integrality gap transference bounds for integer programs

Iskander Aliev, Marcel Celaya, Martin Henk

Abstract

We obtain new transference bounds that connect two active areas of research: proximity and sparsity of solutions to integer programs. Specifically, we study the additive integrality gap of the integer linear programs min{cx: x in P, x integer}, where P={x: Ax=b, x nonnegative} is a polyhedron in the standard form determined by an integer mxn matrix A and an integer vector b. The main result of the paper shows that the integrality gap drops exponentially in the size of support of the optimal solutions that correspond to the vertices of the integer hull of the polyhedron P. Additionally, we obtain a new proximity bound that estimates the distance from any point of P to its nearest integer point in P. The proofs make use of the results from the geometry of numbers and convex geometry.

Sparsity and integrality gap transference bounds for integer programs

Abstract

We obtain new transference bounds that connect two active areas of research: proximity and sparsity of solutions to integer programs. Specifically, we study the additive integrality gap of the integer linear programs min{cx: x in P, x integer}, where P={x: Ax=b, x nonnegative} is a polyhedron in the standard form determined by an integer mxn matrix A and an integer vector b. The main result of the paper shows that the integrality gap drops exponentially in the size of support of the optimal solutions that correspond to the vertices of the integer hull of the polyhedron P. Additionally, we obtain a new proximity bound that estimates the distance from any point of P to its nearest integer point in P. The proofs make use of the results from the geometry of numbers and convex geometry.
Paper Structure (8 sections, 10 theorems, 60 equations)

This paper contains 8 sections, 10 theorems, 60 equations.

Table of Contents

  1. Introduction and main results
  2. Volumes and linear transforms
  3. Proofs of the transference bounds
  4. Proof of Theorem \ref{['thm_transference_gap_integer_hull']}
  5. Proof of Corollary \ref{['cor_gap_immediate']}
  6. Proof of the $\ell_2$-distance proximity bound
  7. Proof of Theorem \ref{['thm:UpperBound']}
  8. Acknowledgement

Key Result

Theorem 1

Let $\boldsymbol A\in \mathbb Z^{m\times n}$ be a matrix of rank $m$, $\boldsymbol b\in \mathbb Z^m$ and $\boldsymbol c\in \mathbb R^n$ be a unit cost vector. Suppose that (initial_IP) is feasible and bounded. Let $\boldsymbol z^*$ be an optimal solution to (initial_IP) which is a vertex of ${P}_I(\ where $s=\| {\boldsymbol z}^*\|_0$.

Theorems & Definitions (16)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Corollary 4
  • lemma 1
  • proof
  • lemma 2
  • proof
  • Corollary 5
  • proof
  • ...and 6 more