Breaking the $2^n$ barrier for directed hamiltonicity

2026-09-30

AI Summary

This study addresses the Hamiltonian cycle problem in directed graphs, which has been constrained by an O*(2^n) time complexity bottleneck since 1962. To overcome this longstanding limitation, this work proposes a novel randomized algorithm that integrates Laplacian determinants, modulo-two counting, and random linearization techniques. Furthermore, it introduces an Isolation Lemma strategy based on arc duplication and deletion to effectively optimize the search space. The primary contribution of this research is achieving an O*(1.9133^n) time complexity, thereby reducing the exponential base for the directed Hamiltonian cycle problem below 1.9133 for the first time. This result breaks a theoretical barrier that has persisted for over sixty years, establishing a new state-of-the-art algorithmic benchmark for this fundamental problem in computational complexity.

Abstract

We give a randomized algorithm for Directed Hamiltonian Cycle on $n$-vertex directed graphs that runs in time $O^*((375/196)^n)=O^*(1.9133^n)$. For general directed graphs, this is the first improvement in the exponential base over the classical $O^*(2^n)$-time algorithms of Bellman and Held--Karp (1962). To obtain this improvement, we first give a $(2-2^{-d})^n \, \text{poly}(n,W)$-time algorithm for counting Hamiltonian paths modulo two at each total weight when at most $d$ distinct weights from $\{1,\ldots,W\}$ enter each vertex. The algorithm combines the Laplacian determinant method of Björklund, Kaski, and Koutis (ICALP 2017) with a random linearization also used by Arvind and Guruswami (IPEC 2021). To apply the isolation lemma while keeping $d$ small, we randomly delete and duplicate arcs, partitioning the incoming copies at each vertex into $d$ groups, where $d\ge2$. We show that, if the input graph has a Hamiltonian path from $s$ to $t$, then with probability at least $\left(1-\frac{1}{1+(2^d-1)^2}\right)^{n-1}$ one can select one group at each vertex other than $s$ so that the selected arcs contain an odd number of such paths.

Problem

• Directed Hamiltonian Cycle
• exponential time complexity
• 2^n barrier
• directed graphs

Innovation

• Directed Hamiltonian Cycle
• Laplacian determinant method
• Random linearization
• Isolation lemma
• Exponential time complexity