On Polynomial Time Approximation Bounded Solution for TSP and NP Complete Problems

TL;DR

This paper explores the possibility that NP equals P by examining polynomial time approximation solutions for the Traveling Salesman Problem in Euclidean space, challenging traditional complexity assumptions.

Contribution

It proposes a novel perspective linking polynomial time approximation solutions of TSP to the broader NP=P question, suggesting a potential proof of NP=P.

Findings

Indicates polynomial time approximation solutions for TSP in Euclidean space.

Suggests that such solutions could imply NP=P.

Provides a new approach to the P vs NP problem.

Abstract

The question of whether all problems in NP class are also in P class is generally considered one of the most important open questions in mathematics and theoretical computer science as it has far-reaching consequences to other problems in mathematics, computer science, biology, philosophy and cryptography. There are intensive research on proving `NP not equal to P' and `NP equals to P'. However, none of the `proved' results is commonly accepted by the research community up to date. In this paper, motived by approximability of traveling salesman problem (TSP) in polynomial time, we aim to provide a new perspective: showing that NP=P from polynomial time approximation-bounded solutions of TSP in Euclidean space.