Idempotent and tropical mathematics. Complexity of algorithms and interval analysis

TL;DR

This paper introduces tropical and idempotent mathematics, highlighting their simplified algorithms and exact interval analysis, especially in linear algebra, contrasting with traditional methods where interval versions are NP-hard.

Contribution

It demonstrates that algorithms in tropical and idempotent mathematics have polynomial complexity and exact interval estimates, unlike traditional linear algebra.

Findings

Interval versions of algorithms have the same complexity as original algorithms.

Linear algebra algorithms over idempotent semirings are polynomial and exact.

Traditional interval linear algebra algorithms are NP-hard and approximate.

Abstract

A very brief introduction to tropical and idempotent mathematics is presented. Tropical mathematics can be treated as a result of a dequantization of the traditional mathematics as the Planck constant tends to zero taking imaginary values. In the framework of idempotent mathematics usually constructions and algorithms are more simple with respect to their traditional analogs. We especially examine algorithms of tropical/idempotent mathematics generated by a collection of basic semiring (or semifield) operations and other "good" operations. Every algorithm of this type has an interval version. The complexity of this interval version coincides with the complexity of the initial algorithm. The interval version of an algorithm of this type gives exact interval estimates for the corresponding output data. Algorithms of linear algebra over idempotent and semirings are examined. In this case, basic algorithms are polynomial as well as their interval versions. This situation is very different from the traditional linear algebra, where basic algorithms are polynomial but the corresponding interval versions are NP-hard and interval estimates are not exact.