On the dynamic compressibility of sets

TL;DR

This paper introduces a new concept of set compressibility based on polynomial dynamics, explores its computational complexity, and discusses lossy compression with open problems for future research.

Contribution

It defines a novel dynamic compressibility measure for sets, reduces the problem to TSP, and establishes NP-completeness results.

Findings

NP-completeness of the compressibility problem

Conditions for $\epsilon$ K-compressibility

Framework for future research and open problems

Abstract

We define a new notion of compressibility of a set of numbers through the dynamics of a polynomial function. We provide approaches to solve the problem by reducing it to the multi-criteria traveling salesman problem through a series of transformations. We then establish computational complexity results by giving some NP-completeness proofs. We also discuss about a notion of $\epsilon$ K-compressibility of a set, with regard to lossy compression and deduce the necessary condition for the given set to be $\epsilon$ K-compressible. Finally, we conclude by providing a list of open problems solutions to which could extend the applicability the our technique.