Linear Programming with Modulo Linear Constraints
If one uses the standard notion of a "linear program", i.e., minimization of a linear functional over a polyhedral set, then it is not clear to me if your problem is actually a linear program. The constraint with the modulo function is equivalent to a disjunction of several affine constraints. This would in general make your problem nonconvex, and thus quite hard.

If you are willing to deal with an integer program, you can exploit the structure. You can write the modulo constraints as  e.g., 
$c_i - c_j -\alpha_{ij} k \geq z_{ij}$, where $\alpha_{ij}$ is an integer for all $i,j$ and a new variable added to your problem. The resulting problem is a mixed integer linear program. Results/algorithms for these are based on the structure of the linear constraints. I am not an expert in MILPs, but I thought this may give you some pointers.