On the Path-Width of Integer Linear Programming

TL;DR

This paper demonstrates that the feasibility of integer linear programming can be decided by representing solutions as graphs with bounded path-width, linking ILP to automata theory and decidability results.

Contribution

It introduces a novel graph-based representation of ILP solutions with bounded path-width, enabling an alternative decidability proof for ILP.

Findings

Solutions can be represented by FO-definable graphs.

One graph per solution has path-width at most 2n.

Decidability of ILP is derived from properties of bounded path-width graphs.

Abstract

We consider the feasibility problem of integer linear programming (ILP). We show that solutions of any ILP instance can be naturally represented by an FO-definable class of graphs. For each solution there may be many graphs representing it. However, one of these graphs is of path-width at most 2n, where n is the number of variables in the instance. Since FO is decidable on graphs of bounded path- width, we obtain an alternative decidability result for ILP. The technique we use underlines a common principle to prove decidability which has previously been employed for automata with auxiliary storage. We also show how this new result links to automata theory and program verification.