ILP Modulo Theories

TL;DR

This paper introduces ILP Modulo Theories, a framework combining Integer Linear Programming with background theories, providing a sound and complete optimization method applicable to various complex problems.

Contribution

It develops the BC(T) transition system, establishing a theoretical foundation for ILP Modulo Theories with proven soundness and completeness.

Findings

BC(T) is sound and complete for decidable, stably-infinite theories.

Prototype BC(T) outperforms leading SMT solvers in tests.

Framework applicable to industrial synthesis, operations research, and software verification.

Abstract

We present Integer Linear Programming (ILP) Modulo Theories (IMT). An IMT instance is an Integer Linear Programming instance, where some symbols have interpretations in background theories. In previous work, the IMT approach has been applied to industrial synthesis and design problems with real-time constraints arising in the development of the Boeing 787. Many other problems ranging from operations research to software verification routinely involve linear constraints and optimization. Thus, a general ILP Modulo Theories framework has the potential to be widely applicable. The logical next step in the development of IMT and the main goal of this paper is to provide theoretical underpinnings. This is accomplished by means of BC(T), the Branch and Cut Modulo T abstract transition system. We show that BC(T) provides a sound and complete optimization procedure for the ILP Modulo T problem, as long as T is a decidable, stably-infinite theory. We compare a prototype of BC(T) against leading SMT solvers.