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

TL;DR

The paper addresses the classical Traveling Salesman Problem by breaking the long-standing $O(2^n n^2)$ dynamic programming barrier. It remodels the Bellman–Held–Karp recurrence as a min-plus matrix product and batches DP rows to push updates, enabling the use of fast min-plus algorithms. The main contribution is a deterministic $2^n n^2 / 2^{\Omega(\sqrt{\log n})}$-time algorithm, derived via a Backbone reduction: if a fast min-plus product for $n\times n$ matrices runs in time $T(n)$, then TSP can be solved in $O(2^n T(n)/n)$; plugging Williams' (and Chan–Williams') subcubic min-plus results yields the claimed bound. This work demonstrates that the classic DP approach is not inherently optimal and opens avenues for further speedups under the min-plus computation framework, with a proof-of-concept implementation available.

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 $2^n n^2 / 2^{Ω(\sqrt{\log n})}$. We achieve this by strategically remodeling the dynamic programming recursion as a min-plus matrix product, for which faster-than-naïve algorithms exist.

Theorems & Definitions (3)

• Theorem 1: Backbone
• Corollary 2: Main
• Theorem 2: Backbone