The Hardness of Approximating Poset Dimension

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Abstract

The dimension of a partially ordered set (poset) is the minimum integer k such that the partial order can be expressed as the intersection of k total orders. We prove that there exists no polynomial-time algorithm to approximate the dimension of a poset on N elements with a factor of O(N0.5ϵ) for any ϵ>0, unless NP=ZPP. The same hardness of approximation holds for the fractional version of poset dimension, which was not even known to be NP-hard.

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