Roughly Polynomial Time: A Concept of Tractability Covering All Known Natural NP-complete Problems

TL;DR

This paper introduces Roughly Polynomial Time (RoughP), a new complexity concept that encompasses all paddable languages, including natural NP-complete problems, by allowing efficient heuristics with exponentially small error rates.

Contribution

The paper defines RoughP, proves it includes all paddable languages, and provides methods to construct encodings and heuristics, bridging practical efficiency and theoretical intractability.

Findings

RoughP includes all paddable languages, covering all known natural NP-complete problems.

Provides a method to construct polynomial-time encodings and errorless heuristics.

Enables efficient generation of large test instances for paddable languages.

Abstract

We introduce a concept of efficiency for which we can prove that it applies to all paddable languages, but still does not conflict with potential worst case intractability. Note that the family of paddable languages apparently includes all known natural NP-complete problems. We call our concept Roughly Polynomial Time (RoughP). A language $L,$ over an at least 2-symbol alphabet, is in RoughP, if the following hold: (1) there exists a bijective encoding $\alpha$ of strings, such that both $\alpha$ and its inverse are computable in polynomial time; (2) there is a polynomial time algorithm $\cal A$, which is an errorless heuristic for $L,$ with exponentially vanishing failure rate relative to the $\alpha$-spheres $S^{(\alpha)}_n=\{\alpha(x)\,|\;\, |x|=n\}$. It means, $\cal A$ always correctly decides whether $x\in L$ or $x\notin L$, whenever it outputs a decision. For some inputs, however, it may not output a decision, rather it may return a special sign, meaning "don't know." But the latter can happen only on an exponentially small fraction of each $\alpha$-sphere. We prove that RoughP contains all paddable languages. This may contribute to the explanation of the often observed gap between practical algorithm performance and theoretical worst case analysis for hard problems. Furthermore, the proof also provides a general method to construct the desired encoding and the errorless heuristic. Additionally, we also show how to use it for efficiently generating large, random, guaranteed positive and negative test instances for any paddable language, including all known natural NP-complete problems. In fact, it appears that every practical decision task (whether in NP or not) can be represented by paddable languages, and, therefore, our RoughP framework applies to all of them. We also explore some connections between RoughP and other complexity classes.