The complexity of string partitioning

Given a string w over a finite alphabet Σ and an integer K, can w be partitioned into strings of length at most K, such that there are no collisions? We refer to this question as the string partition problem and show it is NP-complete for various definitions of collision and for a number of interesting restrictions including |Σ|=2|Σ|=2. This establishes the hardness of an important problem in contemporary synthetic biology, namely, oligo design for gene synthesis.