Efficient diagonalization of symmetric matrices associated with graphs of small treewidth

TL;DR

This paper presents a dynamic programming algorithm for efficiently diagonalizing symmetric matrices associated with graphs of small treewidth, enabling faster computation of determinants, rank, and inertia.

Contribution

The paper introduces a novel algorithm that leverages tree decompositions to diagonalize symmetric matrices efficiently, improving computational complexity for matrices with small treewidth.

Findings

Diagonalization algorithm runs in O(k|T| + k^2 n) time

Allows fast computation of determinant, rank, and inertia

Applicable to matrices over arbitrary fields

Abstract

Let $M=(m_{ij})$ be a symmetric matrix of order $n$ whose elements lie in an arbitrary field $\mathbb{F}$, and let $G$ be the graph with vertex set $\{1,\ldots,n\}$ such that distinct vertices $i$ and $j$ are adjacent if and only if $m_{ij} \neq 0$. We introduce a dynamic programming algorithm that finds a diagonal matrix that is congruent to $M$. If $G$ is given with a tree decomposition $\mathcal{T}$ of width $k$, then this can be done in time $O(k|\mathcal{T}| + k^2 n)$, where $|\mathcal{T}|$ denotes the number of nodes in $\mathcal{T}$. Among other things, this allows one to compute the determinant, the rank and the inertia of a symmetric matrix in time $O(k|\mathcal{T}| + k^2 n)$.