Linear Programming using Limited-Precision Oracles

TL;DR

This paper explores how limited-precision arithmetic oracles can be integrated into linear programming algorithms to compute exact solutions efficiently, bridging practical implementations and theoretical analysis.

Contribution

It introduces the concept of limited-precision LP oracles and demonstrates their effectiveness in computing exact solutions with polynomial complexity.

Findings

A polynomial number of oracle calls suffices for exact LP solutions.

Limited-precision oracles can be integrated into LP algorithms effectively.

The work provides a theoretical foundation for practical LP solving methods.

Abstract

Since the elimination algorithm of Fourier and Motzkin, many different methods have been developed for solving linear programs. When analyzing the time complexity of LP algorithms, it is typically either assumed that calculations are performed exactly and bounds are derived on the number of elementary arithmetic operations necessary, or the cost of all arithmetic operations is considered through a bit-complexity analysis. Yet in practice, implementations typically use limited-precision arithmetic. In this paper we introduce the idea of a limited-precision LP oracle and study how such an oracle could be used within a larger framework to compute exact precision solutions to LPs. Under mild assumptions, it is shown that a polynomial number of calls to such an oracle and a polynomial number of bit operations, is sufficient to compute an exact solution to an LP. This work provides a foundation for understanding and analyzing the behavior of the methods that are currently most effective in practice for solving LPs exactly.