AbstractThe relaxation complexity $${{\,\mathrm{rc}\,}}(X)$$
rc
(
X
)
of the set of integer points X contained in a polyhedron is the smallest number of facets of any polyhedron P such that the integer points in P coincide with X. It is a useful tool to investigate the existence of compact linear descriptions of X. In this article, we derive tight and computable upper bounds on $${{\,\mathrm{rc}\,}}_\mathbb {Q}(X)$$
rc
Q
(
X
)
, a variant of $${{\,\mathrm{rc}\,}}(X)$$
rc
(
X
)
in which the polyhedra P are required to be rational, and we show that $${{\,\mathrm{rc}\,}}(X)$$
rc
(
X
)
can be computed in polynomial time if X is 2-dimensional. Further, we investigate computable lower bounds on $${{\,\mathrm{rc}\,}}(X)$$
rc
(
X
)
with the particular focus on the existence of a finite set $$Y \subseteq \mathbb {Z}^d$$
Y
⊆
Z
d
such that separating X and $$Y \setminus X$$
Y
\
X
allows us to deduce $${{\,\mathrm{rc}\,}}(X) \ge k$$
rc
(
X
)
≥
k
. In particular, we show for some choices of X that no such finite set Y exists to certify the value of $${{\,\mathrm{rc}\,}}(X)$$
rc
(
X
)
, providing a negative answer to a question by Weltge (2015). We also obtain an explicit formula for $${{\,\mathrm{rc}\,}}(X)$$
rc
(
X
)
for specific classes of sets X and present the first practically applicable approach to compute $${{\,\mathrm{rc}\,}}(X)$$
rc
(
X
)
for sets X that admit a finite certificate.
Computational aspects of relaxation complexity: possibilities and limitations
