Ok to do these achievments, it is as follows on PC:
Search Tab, chose under the extras section Has Primary Photo & Search Friends Only then hit search.
Go to last page of the list and work your way back to first page.
Open page for each profile, look for a Pink tab, that same invite them back.
To keep track so you don't open the profile again, rate and like, and at the pime of this post chose Naughty or Nice for the christmas holiday chase.
Easy peasy did 100 in 20min.
At this time this can only be done on pc, and on mobile broswer not on Fubar app.
UPDATE: It seems you get more invite possiblities if you chose to break up the search I described above by female or male.
Have 500+ at the time of posting, and if you go to the same profile again to re-invite wait over 24hrs, 48hrs or so would be best.
More helpful strategies
I use the search like mentioned above, but when you go to the last page the search can only go back 100pages.
With 25 profiles per page, you will only get a max of 2500 and that is if you are lucky that all of them are useful.
So I search using key words, like colors, eyes, lady or other common words used in names.
And I further get them to 2500 or less buy search female or male, friends, with primary picts.
I know there is other blogs out there that describe searching longtime member's friends.
I don't do that not have the time.
I'm around 1300 re-invites (i've been slacking), there are 4 members who have already did all 5k need to for all re-invite achievements.
http://fubar.com/achievements_achievers.php?a=2212
Have fun all ;-)