  The celebrated holographic entanglement entropy triggered investigations on
the connections between quantum information theory and quantum gravity. An
important achievement is that we have gained more insights into the quantum
states. It allows us to diagnose whether a given quantum state is a holographic
state, a state whose bulk dual admits semiclassical geometrical description.
The effective tool of this kind of diagnosis is holographic entropy cone (HEC),
an entropy space bounded by holographic entropy inequalities allowed by the
theory. To fix the HEC and to prove a given holographic entropy inequality, a
proof-by-contraction technique has been developed. This method heavily depends
on a contraction map $f$, which is very difficult to construct especially for
more-region ($n\geq 4$) cases. In this work, we develop a general and effective
rule to rule out most of the cases such that $f$ can be obtained in a
relatively simple way. In addition, we extend the whole framework to relative
homologous entropy, a generalization of holographic entanglement entropy that
is suitable for characterizing the entanglement of mixed states.
