# A Fast Minimum Degree Algorithm and Matching Lower Bound

## TL;DR

This paper introduces a new combinatorial algorithm for exact minimum degree ordering that runs faster than previous methods, provides output-sensitive bounds, and establishes a lower bound under the exponential time hypothesis.

## Contribution

It presents a novel $O(nm)$ time algorithm for exact minimum degree ordering and proves a near-optimal lower bound assuming ETH, advancing understanding of the problem's complexity.

## Key findings

- New $O(nm)$ time algorithm for exact minimum degree ordering
- Output-sensitive bounds for the algorithm's running time
- Proof of a lower bound under ETH for any faster algorithm

## Abstract

The minimum degree algorithm is one of the most widely-used heuristics for reducing the cost of solving large sparse systems of linear equations. It has been studied for nearly half a century and has a rich history of bridging techniques from data structures, graph algorithms, and scientific computing. In this paper, we present a simple but novel combinatorial algorithm for computing an exact minimum degree elimination ordering in $O(nm)$ time, which improves on the best known time complexity of $O(n^3)$ and offers practical improvements for sparse systems with small values of $m$. Our approach leverages a careful amortized analysis, which also allows us to derive output-sensitive bounds for the running time of $O(\min\{m\sqrt{m^+}, \Delta m^+\} \log n)$, where $m^+$ is the number of unique fill edges and original edges that the algorithm encounters and $\Delta$ is the maximum degree of the input graph. Furthermore, we show there cannot exist an exact minimum degree algorithm that runs in $O(nm^{1-\varepsilon})$ time, for any $\varepsilon > 0$, assuming the strong exponential time hypothesis. This fine-grained reduction goes through the orthogonal vectors problem and uses a new low-degree graph construction called $U$-fillers, which act as pathological inputs and cause any minimum degree algorithm to exhibit nearly worst-case performance. With these two results, we nearly characterize the time complexity of computing an exact minimum degree ordering.