Let's break down the process step by step:

Step $1$: Identify Critical Path

First, we need to identify the critical path, which is the longest path through the project network and determines the minimum duration of the project.

Venue Booking $($A$)-2$ days

Vendor Selection $($B$)-3$ days

Event Programming $($E$)-4$ days

Final Preparations $($F$)-3$ days

The critical path is A $->$ B $->$ E $->$ F$,$ with a total duration of $2+3+4+3=12$ days.

Step $2$: Identify Tasks on the Critical Path That Can Be Expedited

Since we need to expedite the project by $2$ days, we must identify tasks on the critical path that can be expedited by $1$ day each.

The tasks on the critical path are B and E$.$