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Graph isomorphism is GI-complete for DAGs: https://en.wikipedia.org/wiki/Graph_isomorphism_problem#Complexity_class_GI. The problem for partial orders is also GI-complete: We can reduce bipartite graph isomorphism (which is GI-complete) to 2 instances of DAG isomorphism where the DAG equals its transitive closure by considering two canonical ways to turn a ...

4

The problem is strongly NP-complete. Reduction from 3-partition, a strongly NP-complete problem. The multiset $S$, $|S| = 3m$, $\sum_{x \in S} x = n$ can be partitioned into tuples of size three of equal sum if and only if $A \leq B$ where $A = S$ and $B$ is $m$ duplicates of $\frac{n}{m}$. If $A \leq B$, we can have $C = A$. If $A \not\leq B$, we must have ...

1

Co-worker here. We haven't solved it yet, but here are a few remarks (in case it gives anyone an idea, because we are stuck). The main thing we have for now is a partial result on so-called crown-free lattices. To show it, for two elements $x,y$ of the poset $P$, I say that $x$ is covered by $y$ if $x\geq y$, $x \neq y$, and there are no elements in between ...

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