 
Summary: Cuts from Proofs:
A Complete and Practical Technique for Solving
Linear Inequalities over Integers
Isil Dillig Thomas Dillig Alex Aiken
{isil, tdillig, aiken}@cs.stanford.edu
Department of Computer Science
Stanford University
Abstract. We propose a novel, sound, and complete Simplexbased al
gorithm for solving linear inequalities over integers. Our algorithm, which
can be viewed as a semantic generalization of the branchandbound tech
nique, systematically discovers and excludes entire subspaces of the so
lution space containing no integer points. Our main insight is that by
focusing on the defining constraints of a vertex, we can compute a proof
of unsatisfiability for the intersection of the defining constraints and use
this proof to systematically exclude subspaces of the feasible region with
no integer points. We show experimentally that our technique signifi
cantly outperforms the top four competitors in the QFLIA category of
the SMTCOMP '08 when solving linear inequalities over integers.
1 Introduction
A quantifierfree system of linear inequalities over integers is defined by Ax b
