Clifford Algebras Meet Tree Decompositions.

Area of Science:

• * Theoretical Computer Science
• * Algebra
• * Graph Theory

Background:

• * Determinant-based algorithms often involve complex convolutions.
• * Clifford algebras, generalizations of quaternions, offer advanced mathematical tools.
• * Efficient subgraph counting is crucial in various computational problems.

Purpose of the Study:

• * To introduce a novel non-commutative subset convolution for determinant-based algorithms.
• * To leverage Clifford algebras for computational efficiency.
• * To enhance algorithms for counting graph subgraphs parameterized by treewidth.

Main Methods:

• * Development of a non-commutative subset convolution.
• * Application of Clifford algebras for computational speed-up.
• * Design of new algorithms for subgraph counting problems.

Main Results:

• * An improved time complexity for counting Steiner trees.
• * An improved time complexity for counting Hamiltonian cycles.
• * Achieved best-known deterministic running times for related decision problems.

Conclusions:

• * The non-commutative subset convolution is a powerful tool for specific algorithmic challenges.
• * Clifford algebras provide an effective framework for optimizing these computations.
• * The new algorithms offer significant performance improvements over existing methods.