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Generalized sorting algorithm on partially ordered set generated by a relation

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Problem Detail: 

Assume we have a finite set $X$ of elements and any relation $\preceq$ on $X$. Such a relation may or may not generate a reflexive transitive anti-symmetric relation $\leq$ on $X$ (a partial order). Recall:

reflexive: We always have $x\leq x$.

transitive: If $x\leq y$ and $y\leq z$ then also $x\leq z$.

anti-symmetric: If $x\leq y$ and $y\leq x$ then $x=y$.

I'm looking for an efficient algorithm that decides whether $\preceq$ extends to a partial order and if yes, returns one or one with uniform probability or all possible orderings of $X$ which respect the partial order.

This naturally decomposes into two subtasks (not claiming that the best algorithm decomposes this way):

  1. Decision whether $\preceq$ extends to $\leq$ or, in other words, extending by transitivity does not violate anti-symmetry. If it does, describing the partial order $\leq$.

  2. Finding the possible orderings.

The second points is related this question.

Asked By : Werner Thumann

Answered By : D.W.

You can efficiently test whether $\preceq$ extends to a partial order using topological sorting. Form a directed graph with vertex set $X$ and with an edge $x \to y$ whenever we have $x \preceq y$ and $x \ne y$. Then $\preceq$ extends to a partial order if and only if this graph has no cycles, which can be checked in linear time (e.g., by topologically sorting the graph).

Moreover, we can describe a minimal partial order $\le$ that is consistent with $\preceq$, as follows. Compute the transitive closure of this graph. Now declare $x \le y$ iff there's an edge $x \to y$ in the transitive closure, or $x=y$. (Equivalently, declare $x \le y$ iff $y$ is reachable from $x$ in zero or more steps, in the graph derived from $\preceq$.) By construction, the relation $\le$ defined in this way will automatically be reflexive and transitive; if the original graph has no cycles, then it is easy to prove that the relation $\le$ will be anti-symmetric as well and thus will be a partial order.

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