  Characterizing the solution sets in a problem by closedness under operations
is recognized as one of the key aspects of algorithm development, especially in
constraint satisfaction. An example from the Boolean satisfiability problem is
that the solution set of a Horn conjunctive normal form (CNF) is closed under
the minimum operation, and this property implies that minimizing a nonnegative
linear function over a Horn CNF can be done in polynomial time. In this paper,
we focus on the set of integer points (vectors) in a polyhedron, and study the
relation between these sets and closedness under operations from the viewpoint
of 2-decomposability. By adding further conditions to the 2-decomposable
polyhedra, we show that important classes of sets of integer vectors in
polyhedra are characterized by 2-decomposability and closedness under certain
operations, and in some classes, by closedness under operations alone. The most
prominent result we show is that the set of integer vectors in a
unit-two-variable-per-inequality polyhedron can be characterized by closedness
under the median and directed discrete midpoint operations, each of these
operations was independently considered in constraint satisfaction and discrete
convex analysis.
