Suppose we have a convex hull computed as the solution to a linear programming problem (via whatever method you want). Given this convex hull (and the inequalities that formed the convex hull) is there a fast way to compute the integer points on the surface of the convex hull? Or is the problem NP?


There exist ways to bound the number of integer points and to find the number of integer points inside convex hull, but I specifically want the points on the hull itself.


EDIT: Suppose the set of inequalities (the linear program) have integer coefficients /EDIT



Finding integer points on an N-d convex hull
Because the facets of your convex hull are themselves polytopes (of one lower dimension—$d{=}21$ in your case), it seems your question is equivalent to asking how to count lattice points in a polytope. One paper on this topic is "[The Many Aspects of Counting Lattice Points in Polytopes](http://www.math.ucdavis.edu/~deloera/RECENT_WORK/semesterberichte.pdf)" by J. A. DeLoera. Section 4 is entitled, "Actually Counting: Modern Algorithms and Software." A key reference in that section is to a paper by Barvinok entitled, "Polynomial time algorithm for counting integral points in 
polyhedra when the dimension is ﬁxed" (*Math of Operations Research*, Vol. 19, 769–779, 1994), whose title (*polynomial time*) seems to provide an answer of sorts to your question.


