QUBO formulations for NP-Hard spanning tree problems

TL;DR

This paper presents a new QUBO formulation for spanning tree problems that encodes spanning trees as permutations, reducing variable complexity for certain NP-hard variants.

Contribution

It introduces a permutation-based QUBO formulation for spanning trees, improving variable efficiency for the k-minimum spanning tree problem.

Findings

Reduced variable count to $\\mathcal{O}(|V|k)$ for k-minimum spanning tree

Applied formulation to four NP-hard spanning tree variants

Demonstrated efficiency over previous strategies

Abstract

We introduce a novel Quadratic Unconstrained Binary Optimization (QUBO) formulation method for spanning tree problems. Instead of encoding the presence of edges in the tree individually, we opt to encode spanning trees as a permutation problem. We apply our method to four NP-hard spanning tree variants, namely the k-minimum spanning tree, degree-constrained minimum spanning tree, minimum leaf spanning tree, and maximum leaf spanning tree. Our main result is a formulation with $\mathcal{O}(|V|k)$ variables for the k-minimum spanning tree problem, beating related strategies that need $\mathcal{O}(|V|^{2})$ variables.