The Complexity of the Partial Order Dimension Problem: Closing the Gap

Abstract

The dimension of a partial order P is the minimum number of linear orders whose intersection is P . There are efficient algorithms to test if a partial order has dimension at most 2. In 1982 Yannakakis [25] showed that for k ≥ 3 to test if a partial order has dimension ≤ k is NP-complete. The height of a partial order P is the maximum size of a chain in P . Yannakakis also showed that for k ≥ 4 to test if a partial order of height 2 has dimension ≤ k is NP-complete. The complexity of deciding whether an order of height 2 has dimension 3 was left open. This question became one of the best known open problems in dimension theory for partial orders. We show that the problem is NP-complete.Technically, we show that the decision problem (3DH2) for dimension is equivalent to deciding for the existence of bipartite triangle containment representations (BTCon). This problem then allows a reduction from a class of planar satisfiability problems (P-3-CON-3-SAT(4)) which is known to be NP-hard.Mathematics Subject Classifications (2010) 06A07, 68Q25, 05C62,

What Makes the Recognition Problem Hard for Classes Related to Segment and String graphs?

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Sublinear approximation algorithms for boxicity and related problems

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