Using a single verb, I want to express the act of making something negligible, i.e. very small or insignificant. "Neglate" doesn't seem to be english :)
About the delay between two events in time. It should be possible to almost
<word here> this delay.
Second example, as some people didn't get what I intended.
About cutting the cost of something. It should be possible to
<word here> the cost. And no, not minimize, it is really about cutting it all the way down to almost nothing, to something that is negligible.