An efficient tree decomposition method for permanents and mixed discriminants

TL;DR

This paper introduces an efficient algorithm for computing permanents, mixed discriminants, and hyperdeterminants of structured matrices and tensors, leveraging small treewidth to improve computational complexity.

Contribution

It presents a novel algorithm that exploits the sparsity and treewidth of matrices and tensors to compute complex invariants more efficiently.

Findings

Algorithm requires $O(n 2^)$ operations for permanents

Computes mixed discriminants and hyperdeterminants with $O(n^2 + n 3^)$ complexity

Shows mixed volume computation remains hard even with bounded treewidth

Abstract

We present an efficient algorithm to compute permanents, mixed discriminants and hyperdeterminants of structured matrices and multidimensional arrays (tensors). We describe the sparsity structure of an array in terms of a graph, and we assume that its treewidth, denoted as $\omega$, is small. Our algorithm requires $O(n 2^\omega)$ arithmetic operations to compute permanents, and $O(n^2 + n 3^\omega)$ for mixed discriminants and hyperdeterminants. We finally show that mixed volume computation continues to be hard under bounded treewidth assumptions.