Efficient programs of NPC problems should be length upper-bounded, and a thought experiment to search for them by machine enumeration

TL;DR

This paper explores the theoretical limits of efficient algorithms for NP-complete problems, proposing a framework for identifying bounded programs with polynomial time complexity through machine enumeration and complexity measures.

Contribution

It introduces the concept of bounded algorithms, a length upper bound for efficient programs, and a growth rate characteristic function for complexity evaluation, advancing the theoretical understanding of NPC problem algorithms.

Findings

Defines execution time as loading plus running time on Turing Machines

Proposes a criterion to identify inefficient bounded algorithms

Raises the question of polynomial bounded algorithms for NPC problems

Abstract

This paper proposes a thought experiment to search for efficient bounded algorithms of NPC problems by machine enumeration. The key contributions are: -- On Universal Turing Machines, a program's time complexity should be characterized as: execution time(n) = loading time(n) + running time(n). -- Introduces the concept of bounded algorithms; proposes a comparison based criterion to decide if a bounded algorithm is inefficient; and establishes the length upper bound of efficient bounded programs. -- Introduces the growth rate characteristic function to evaluate program complexity, which is more easily machine checkable based on observations. -- Raises the theoretical question: if there exists any bounded algorithm with polynomial execution time for NPC problems.