How can I expire a cookie?

I am in the middle of conducting a poll and I want my voters to vote again after 12 hours. My voters can't vote again. I tried turning off IP checking, but that doesn't work. I don't want to ask my visitors to delete thier cookies. For the users that want to vote over again, how can I get them to vote without deleting their cookies. Turning off IP checking does not work.

Is there a way for me to expire there cookies and allow them to vote again

Thanks anyone for your help

edit the cookie code snippet (poll_cookie.php)

<?php
.
.
$cookie_expire = 12; // hours
.
.
?>