Clifford algebras meet tree decompositions

TL;DR

This paper introduces a novel convolution technique using Clifford algebras to improve deterministic algorithms for counting specific subgraphs like Steiner trees and Hamiltonian cycles, parameterized by treewidth.

Contribution

It presents the Non-commutative Subset Convolution and leverages Clifford algebras to achieve faster algorithms for graph problems parameterized by treewidth.

Findings

Improved algorithms for counting Steiner trees and Hamiltonian cycles.

New deterministic algorithms matching the best known bounds under certain assumptions.

Application to a deterministic Feedback Vertex Set algorithm.

Abstract

We introduce the Non-commutative Subset Convolution - a convolution of functions useful when working with determinant-based algorithms. In order to compute it efficiently, we take advantage of Clifford algebras, a generalization of quaternions used mainly in the quantum field theory. We apply this tool to speed up algorithms counting subgraphs parameterized by the treewidth of a graph. We present an $O^*((2^\omega + 1)^{tw})$-time algorithm for counting Steiner trees and an $O^*((2^\omega + 2)^{tw})$-time algorithm for counting Hamiltonian cycles, both of which improve the previously known upper bounds. The result for Steiner Tree also translates into a deterministic algorithm for Feedback Vertex Set. All of these constitute the best known running times of deterministic algorithms for decision versions of these problems and they match the best obtained running times for pathwidth parameterization under assumption $\omega = 2$.