Solving Integer Linear Programs with a Small Number of Global Variables and Constraints

TL;DR

This paper introduces fracture backdoors, a structural measure for ILP instances, and develops algorithms and complexity bounds for solving ILPs with few global variables or constraints, enhancing understanding of ILP tractability.

Contribution

The paper formalizes fracture backdoors, provides algorithms to compute them, and develops parameterized algorithms with matching lower bounds for ILP instances.

Findings

Exact and approximation algorithms for fracture backdoors.

Parameterized algorithms for ILP based on global variables/constraints.

Matching lower bounds for ILP complexity with fracture backdoors.

Abstract

Integer Linear Programming (ILP) has a broad range of applications in various areas of artificial intelligence. Yet in spite of recent advances, we still lack a thorough understanding of which structural restrictions make ILP tractable. Here we study ILP instances consisting of a small number of "global" variables and/or constraints such that the remaining part of the instance consists of small and otherwise independent components; this is captured in terms of a structural measure we call fracture backdoors which generalizes, for instance, the well-studied class of N -fold ILP instances. Our main contributions can be divided into three parts. First, we formally develop fracture backdoors and obtain exact and approximation algorithms for computing these. Second, we exploit these backdoors to develop several new parameterized algorithms for ILP; the performance of these algorithms will naturally scale based on the number of global variables or constraints in the instance. Finally, we complement the developed algorithms with matching lower bounds. Altogether, our results paint a near-complete complexity landscape of ILP with respect to fracture backdoors.