On Polynomial Kernels for Integer Linear Programs: Covering, Packing and Feasibility

TL;DR

This paper investigates the limits of polynomial kernelization for integer linear programs, showing non-existence results for feasibility problems and providing a polynomial kernel for a specific covering ILP case with row-sparse matrices.

Contribution

It establishes non-existence of polynomial kernels for ILP feasibility under certain parameters and introduces a polynomial kernel for row-sparse covering ILPs.

Findings

ILP feasibility has no polynomial kernel unless NP ⊆ coNP/poly.

Polynomial kernel exists for row-sparse Cover ILP when parameterized by solution size.

Results extend to bounded variable degree and packing ILPs.

Abstract

We study the existence of polynomial kernels for the problem of deciding feasibility of integer linear programs (ILPs), and for finding good solutions for covering and packing ILPs. Our main results are as follows: First, we show that the ILP Feasibility problem admits no polynomial kernelization when parameterized by both the number of variables and the number of constraints, unless NP \subseteq coNP/poly. This extends to the restricted cases of bounded variable degree and bounded number of variables per constraint, and to covering and packing ILPs. Second, we give a polynomial kernelization for the Cover ILP problem, asking for a solution to Ax >= b with c^Tx <= k, parameterized by k, when A is row-sparse; this generalizes a known polynomial kernelization for the special case with 0/1-variables and coefficients (d-Hitting Set).