I am having the exact same problem. Whenever i try to start Hitman: Absolution (Steam), i am instantly greeted by an error message: "Failed to initialize DirectX 11".
First of all, my system meets all recommended system specs. I have windows vista 64-bit, everything updated. I also have the newest version of DirectX.
I have tried several updates to my video drivers, windows service packs, etc. etc. but nothing has worked so far. I also tried using a no-cd patch (i'm not trying to endorse Piracy here!), to see if Steam itself was the problem, but that results in even weirder error messages. A game cache integrity check did not solve the problem. Also sometimes, instead of the DirectX error, i get a "Steam servers too busy" error.
If somebody else could share their experience with this problem, or even tell everybody how to fix it, i would be more than grateful.