Well for one, I'm guessing by "paths" you actually mean URLs. To that end, note that using relative URLs (e.g. anything that doesn't begin with the protocol, such as 'http://mysite.com/foo') is not currently allowed (though I've heard that might change in the (near?) future).
For another, it's not the server that's opening any connections at all. It's the client.
Finally, even if you allow Keep-Alive connections, it's unlikely that using an absolute URL will force the browser to close the connection and try to create a new one. I'll add one caveat - I'm assuming that your absolute URLs use the same protocol and hostname as that which the client used to make the initial request. In other words, you aren't redirecting them to www.mysite.com after they first arrived at mysite.com (since the two are separate, unique hostnames and thus deserve to be treated as such).
What's more, even if that really were the case, that doesn't justify (IMHO) why that's the source of the problem. If your hosting company can't handle the shape and/or volume of your website's traffic, it doesn't make much sense that the culprit is that you aren't using an HTTP header in a way that would violate the current HTTP standard (albeit in a way that most modern browsers will still be able to handle).