### Abstract:

The set of polyhedral pointed rational cones form a partially ordered set with
respect to elementary additive extensions of certain type. This poset captures
a global picture of the interaction of all possible rational cones with the integer
lattice and, also, provides an alternative approach- to another important poset,
the poset of normal polytopes. One of the central conjectures in the field, is
the s.c. Cone Conjecture: the order on cones is the inclusion order. The conjecture
has been proved only in dimensions up to 3. In this work we develop
an algorithmic approach to the conjecture in higher dimensions. Namely, we
study how often two specific types of cone extensions generate the chains of
cones in dimensions 4 and 5, whose existence follows from the Cone Conjecture.
A naive expectation, explicitly expressed in a recently published paper, is that
these special extensions suffice to generate the desired chains. This would prove
the conjecture in general and was the basis of the proof of the 3-dimesional case.
Our extensive computational experiments show that in many cases the desired
chains are in fact generated, but there are cases when the chain generation process
does not terminate in reasonable time. Moreover, the fast generation of the
desired chains fails in an interesting way - the complexity of the involved cones,
measured by the size of their Hilbert bases, grows roughly linearly in time, making
it less and less likely that we have a terminating process. This phenomenon
is not observed in dimension 3. Our computations can be done in arbitrary high
dimensions. We make a heavy use of SAGE, an open-source mathematics software
system, and Normaliz, a C++ package designed to compute the Hilbert
bases of cones.