[helper]: The role of \omega

[atlasjunjay]

I have problems in formula derivation. Very disturbing! Thank you very much!
In Section II. B, the authors state “Note that, due to the relaxation variable \omega, enforcing multiple DCBF constraints for each hi is equivalent to enforcing a single DCBF constraint.” I am confused about this issue. Why the relaxation variable \omega leads to this case? Is the statement not true without the relaxation variable \omega?
In Section III. C, the authors state “Additionally, we can introduce the relaxation
variables without affecting the analysis in this section.” Why?