How relevant is LHC to quantum gravity? Premise: the LHC is obviously mapping unseen territory in high energies, and therefore it's always possible to imagine far out results.
Excluding completely unexpected outcomes - is the LHC performing any experiment that could help with string theory or m-theory? For example:


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*direct super-strings or m-theory predictions to be tested or confuted


but also


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*measurements that would help "shape" string/m-theory into something more concretely testable or practical than the current blurry incarnation?

 A: This article from CERN Courier is about string-theory and experimental tests of it, and about LHC. It is not really technical but it also links to various references and articles.
And for more technical info, you can browse the list of String Theory Seminar of the TH department at CERN and search for "LHC"; very interesting.
A: Well, there's no reason to believe in supersymmetry, beyond some theoretical niceness to it, so if they see THAT at the LHC, then string theory gets a big boost, as there is no way other than supersymmetry to produce fermions in string theory.
The other thing that might be relevant to quantum gravity is that if there are large extra dimensions (as in, large compared to the Planck length, but smaller than detectable by things like the Cavendish experiment).  If that is the case, then the 'fundamental' gravitational constant may be much larger than Newton's constant (they differ by a factor of the volume of the large extra dimensions), and quantum gravitational effects would be accessible at the LHC.  
