A Reduction from Unbounded Linear Mixed Arithmetic Problems into Bounded Problems

TL;DR

This paper introduces a polynomial-time method combining transformations to convert unbounded linear mixed arithmetic problems into bounded ones, enabling more efficient satisfiability checking without precomputing variable bounds.

Contribution

The paper presents a novel combination of transformations that preserve satisfiability and convert unbounded problems into bounded ones, improving efficiency over existing methods.

Findings

Transformations preserve satisfiability and can be computed in polynomial time.

Bounded systems facilitate termination in existing solving approaches.

Experimental results demonstrate practical efficiency of the transformations.

Abstract

We present a combination of the Mixed-Echelon-Hermite transformation and the Double-Bounded Reduction for systems of linear mixed arithmetic that preserve satisfiability and can be computed in polynomial time. Together, the two transformations turn any system of linear mixed constraints into a bounded system, i.e., a system for which termination can be achieved easily. Existing approaches for linear mixed arithmetic, e.g., branch-and-bound and cuts from proofs, only explore a finite search space after application of our two transformations. Instead of generating a priori bounds for the variables, e.g., as suggested by Papadimitriou, unbounded variables are eliminated through the two transformations. The transformations orient themselves on the structure of an input system instead of computing a priori (over-)approximations out of the available constants. Experiments provide further evidence to the efficiency of the transformations in practice. We also present a polynomial method for converting certificates of (un)satisfiability from the transformed to the original system.