TSP Escapes the $O(2^n n^2)$ Curse

TL;DR

This paper presents a groundbreaking deterministic algorithm for the Traveling Salesman Problem that surpasses the long-standing $O(2^n n^2)$ time barrier by leveraging advanced matrix multiplication techniques.

Contribution

The authors introduce a novel approach that remodels the dynamic programming recursion as a min-plus matrix product, achieving faster-than-naive algorithms.

Findings

Breaks the $O(2^n n^2)$ barrier for TSP.

Achieves a runtime of $2^n n^2 / 2^{ ext{Omega}(\sqrt{ ext{log} n})}$.

Demonstrates the first improvement in deterministic TSP algorithms in over sixty years.

Abstract

The dynamic programming solution to the traveling salesman problem due to Bellman, and independently Held and Karp, runs in time $O(2^n n^2)$, with no improvement in the last sixty years. We break this barrier for the first time by designing an algorithm that runs in deterministic time $2^n n^2 / 2^{\Omega(\sqrt{\log n})}$. We achieve this by strategically remodeling the dynamic programming recursion as a min-plus matrix product, for which faster-than-na\"ive algorithms exist.