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

A Fast Minimum Degree Algorithm and Matching Lower Bound