Quantum speedups for treewidth

TL;DR

This paper introduces three quantum algorithms that significantly accelerate the computation of graph treewidth compared to classical methods, leveraging quantum search and dynamic programming techniques.

Contribution

It presents the first quantum algorithms for exact treewidth computation with exponential speedups over classical algorithms, using QRAM and quantum techniques.

Findings

Quantum algorithms achieve up to 1.538^n time complexity.

Classical algorithm complexity is 1.755^n, quantum algorithms improve this.

New classical time-space tradeoff for treewidth in 2^n time and sqrt(2^n) space.

Abstract

In this paper, we study quantum algorithms for computing the exact value of the treewidth of a graph. Our algorithms are based on the classical algorithm by Fomin and Villanger (Combinatorica 32, 2012) that uses $O(2.616^n)$ time and polynomial space. We show three quantum algorithms with the following complexity, using QRAM in both exponential space algorithms: $\bullet$ $O(1.618^n)$ time and polynomial space; $\bullet$ $O(1.554^n)$ time and $O(1.452^n)$ space; $\bullet$ $O(1.538^n)$ time and space. In contrast, the fastest known classical algorithm for treewidth uses $O(1.755^n)$ time and space. The first two speed-ups are obtained in a fairly straightforward way. The first version uses additionally only Grover's search and provides a quadratic speedup. The second speedup is more time-efficient and uses both Grover's search and the quantum exponential dynamic programming by Ambainis et al. (SODA '19). The third version uses the specific properties of the classical algorithm and treewidth, with a modified version of the quantum dynamic programming on the hypercube. Lastly, as a small side result, we also give a new classical time-space tradeoff for computing treewidth in $O^*(2^n)$ time and $O^*(\sqrt{2^n})$ space.