The Complexity of the Extendibility Problem for Finite Posets

For a finite poset P let EXT(P ) denote the following decision problem. Given a finite poset Q and a partial map f from Q to P , decide whether f extends to a monotone total map from Q to P . It is easy to see that EXT(P ) is in the complexity class NP. In [SIAM J. Comput., 28 (1998), pp. 57–104], Feder and Vardi define the classes of width 1 and of bounded strict width constraint satisfaction problems for finite relational structures. Both classes belong to the broader class of bounded width problems in P. We prove that for any finite poset P , if EXT(P ) has bounded strict width, then it has width 1. In other words, if a poset admits a near unanimity operation, it also admits a totally symmetric idempotent operation of any arity. In [Fund. Inform., 28 (1996), pp. 165–182], Pratt and Tiuryn proved that SAT(P ), a polynomial-time equivalent of EXT(P ) is NPcomplete if P is a crown. We generalize Pratt and Tiuryn’s result on crowns by proving that EXT(P ), is NP-complete for any finite poset P which admits no nontrivial idempotent Malcev condition.