The Complexity Landscape of Decompositional Parameters for ILP

TL;DR

This paper explores the complexity of Integer Linear Programming (ILP) by analyzing how structural parameters of the constraint matrix affect tractability, identifying new fixed-parameter tractable cases and providing a comprehensive complexity overview.

Contribution

It introduces new tractable fragments of ILP based on treedepth and coefficient bounds, expanding understanding beyond classical total unimodularity.

Findings

ILP is fixed-parameter tractable when parameterized by treedepth and maximum coefficient.

Hardness results show ILP remains complex when parameterized by treewidth.

Provides a comprehensive complexity landscape of ILP based on decompositional parameters.

Abstract

Integer Linear Programming (ILP) can be seen as the archetypical problem for NP-complete optimization problems, and a wide range of problems in artificial intelligence are solved in practice via a translation to ILP. Despite its huge range of applications, only few tractable fragments of ILP are known, probably the most prominent of which is based on the notion of total unimodularity. Using entirely different techniques, we identify new tractable fragments of ILP by studying structural parameterizations of the constraint matrix within the framework of parameterized complexity. In particular, we show that ILP is fixed-parameter tractable when parameterized by the treedepth of the constraint matrix and the maximum absolute value of any coefficient occurring in the ILP instance. Together with matching hardness results for the more general parameter treewidth, we give an overview of the complexity of ILP w.r.t. decompositional parameters defined on the constraint matrix.