  The orientation completion problem for a fixed class of oriented graphs asks
whether a given partially oriented graph can be completed to an oriented graph
in the class. Orientation completion problems have been studied recently for
several classes of oriented graphs, yielding both polynomial time solutions and
NP-completeness results. Local tournaments are a well-structured class of
oriented graphs that generalize tournaments and their underlying graphs are
intimately related to proper circular-arc graphs. Proper interval graphs are
precisely those which can be oriented as acyclic local tournaments. It has been
proved that the orientation completion problems for local tournaments and
acyclic local tournaments are both polynomial time solvable. In this paper we
identify the obstructions for acyclic local tournament orientation completions.
These are in a sense minimal partially oriented graphs that cannot be completed
to acyclic local tournaments. Our description of the obstructions imply that
they can be recognized in polynomial time. In a companion paper we will
determine all obstructions for local tournament orientation completions.
