Open source login based wiki

I suggest trying mediawiki, this is the software that wikipedia runs on.

An addon called Approved Revs is available for approving revisions.

"Approved Revs is an extension that lets administrators mark a certain revision of a page as "approved". The approved revision is the one displayed when users view the page at its main URL."

A more complex revision management addon, Flagged Revisions, is also available.

Possible solution for DokuWiki

A post in the DokuWiki User Forum "Relative links in wikipages" mentions this tag.

It gives a path to the wiki root, so a link like this one [[this>../../VolD/Projects/]] can link to some page outside the wiki folder without using a full URL.

See also the official documentation: https://www.dokuwiki.org/interwiki

The rest of this answer was written before I knew that the question was about DokuWiki. I'll leave it for future reference.

The problem if you're using MediaWiki

In a wiki, it can be ambiguous to use relative paths because it is common to rewrite URLs. Let's take Wikipedia as an example. When you access http://en.wikipedia.org/wiki/Main_Page, you're not looking at a document called Main_Page in /wiki. Instead, it is /w/index.php that does the work, as you can see from http://en.wikipedia.org/w/index.php?title=Main_Page returning the same output. So, relative URLs will introduce an ugly ambiguity here. Relative from where? (See also http://yoast.com/relative-urls-issues/) As the rewritten URL can be anything, it won't be smart to rely on relative paths.

Workaround for MediaWiki

I suspect that the reason you want to use a relative path is because you want to link to a file on the same server, but is not managed within the wiki's database. In that case, you can use the magic word {{SERVERNAME}}.

[http://{{SERVERNAME}}/rel/path Link text]

should give you the link, with {{SERVERNAME}} replaced accordingly (see here). Taking Wikipedia as an example, {{SERVERNAME}} = "en.wikipedia.org".

So this is not exactly a relative path, but playing around with other magic words such as {{SCRIPTPATH}} and {{STYLEPATH}} might offer you a solution, depending on what directory you want to access. (For the Wikipedia example, {{SCRIPTPATH}} = "/w".) See http://www.mediawiki.org/wiki/Help:Magic_words#Technical_metadata