Dyadic linear programming and extensions

TL;DR

This paper investigates the problem of finding dyadic optimal solutions in linear programming, providing polynomial-time algorithms, bounds on solution size, and extending the framework beyond dyadic cases.

Contribution

It introduces polynomial-time methods for dyadic linear programs, establishes bounds on solutions, and identifies key properties enabling these solutions.

Findings

Polynomial-time algorithms for dyadic LPs.

Bounds on support size and denominators.

Extension of framework beyond dyadic case.

Abstract

A rational number is dyadic if it has a finite binary representation $p/2^k$, where $p$ is an integer and $k$ is a nonnegative integer. Dyadic rationals are important for numerical computations because they have an exact representation in floating-point arithmetic on a computer. A vector is dyadic if all its entries are dyadic rationals. We study the problem of finding a dyadic optimal solution to a linear program, if one exists. We show how to solve dyadic linear programs in polynomial time. We give bounds on the size of the support of a solution as well as on the size of the denominators. We identify properties that make the solution of dyadic linear programs possible: closure under addition and negation, and density, and we extend the algorithmic framework beyond the dyadic case.