Discussion
Why is finding an optimal elimination order hard?
Answer: D
Choosing an order minimising the largest factor is an NP hard problem, so heuristics such as minimum degree or minimum fill are used. The numbers are irrelevant since only the structure matters, orders differ enormously in cost, and one order may serve many queries. That the preprocessing step is itself intractable, and is done heuristically, is a common situation and does not prevent the main algorithm from being useful. Naming the cost of a choice alongside its benefit is what turns a fact into a design decision.
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