## Can two admissable heuristics not dominate each other?

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I am working on a project for my artificial intelligence class. I was wondering if I have 2 admissible heuristics, A and B, is it possible that A does not dominate B and B does not dominate A? I am wondering this because I had to prove if each heuristic is admissible and I did that, and then for each admissible heuristic, we have to prove if each one dominates the other or not. I think I have a case that neither dominates the other and I was wondering if maybe I got the admissibility wrong because of that.

Specifically, you may find that sometimes $$h_1 < h_2$$ and in other times $$h_2 < h_1$$, where $$h_1$$ and $$h_2$$ are admissible heuristics. Thus, by definition, neither strictly dominates the other.
$$h_3 = \max(h_1, h_2)$$
1I would like to note that $\max(h_1, h_2)$ gives you the best of both $h_1$ and $h_2$, if $h_1$ and $h_2$ are admissible: the idea is that, by taking the maximum of both, they are closer to the optimal heuristic. You hadn't explicitly stated that $h_1$ and $h_2$ are admissible. I edited your answer to add this detail. – nbro – 2019-11-11T15:46:27.847