Solving less-than inequalities with positive integers

Pyth, 39 bytes

V>1f.A<MxMLTN.pS{s=Nm%2cd).zVNjd[H\==hZ

Try it online: Demonstration

Brute-forces through all possible permutations (and interpret them as sortings), check if they match the inequalities, and assign them the values 1, 2, ...., n.

Explanation

f.A<MxMLTN.pS{s=Nm%2cd).z  
                 m     .z  map each input line d to:
                    cd)       split d by spaces
                  %2          and remove the second element
               =N          save this list of pairs to N
              s            combine these pairs to a big list of variable names
             {             set (remove duplicates)
          .pS              generate all permutations
f                          filter for permutations T, which satisfy:
     xMLTN                    replace each variable in N by their index in T
 .A<M                         check if each pair is ascending

V>1...VNjd[H\==hZ          implicit: Z = 0
 >1                        remove all but the last filtered permutation (if any)
V                          for each permutation N in ^ (runs zero times or once):
      VN                      for each variable H in N:
          [                      generate a list containing:
           H                        H
            \=                      "="
              =hZ                   Z incremented by 1 (and update Z)
        jd                       join this list by spaces and print

CJam (53 52 49 bytes)

qS-N/'<f/:A:|e!{A{1$1$&=!},!*},:ee{()" = "\N}f%1<

Online demo

This brute-forces all permutations of the distinct tokens, filtering for those assignments of the numbers 0 to n-1 which obey all of the constraints, and then formats them, incrementing the numbers, and presents the first one. This is certain to find a solution if there is one, because it's essentially a topological sort.

Thanks to Reto Koradi for 3 chars and Martin Büttner for 1.

Mathematica, 83 bytes

Quiet@Check[Equal@@@FindInstance[Join[#,#>0&/@(v=Sequence@@@#)],v,Integers][[1]],]&

Takes input as a list of inequalities. Either outputs a list of assignments or Null if it is impossible.