Approximating 1-dimensional TSP Requires Omega(n log n) Comparisons

TL;DR

This paper proves that any comparison-based algorithm approximating the 1-dimensional TSP within a certain factor must perform at least on the order of n log n comparisons, establishing a fundamental complexity limit.

Contribution

It provides a concise proof that approximating 1D TSP within a specific ratio inherently requires Omega(n log n) comparisons, highlighting a key complexity barrier.

Findings

Comparison-based n^(1-epsilon)-approximation algorithms need Omega(n log n) comparisons

Establishes a lower bound on the complexity of 1D TSP approximation

Shows fundamental limitations for comparison-based approaches

Abstract

We give a short proof that any comparison-based n^(1-epsilon)-approximation algorithm for the 1-dimensional Traveling Salesman Problem (TSP) requires Omega(n log n) comparisons.