Spatial codes and the hardness of string folding problems

We present the first proof of NP-hardness (under randomized polynomial time reductions) for string folding problems over a finite alphabet. All previous such intractability results have required an unbounded alphabet size. These problems correspond to the protein folding problem in variants of the hydrophobic-hydrophilic (or HP) model with a fixed number of monomer types. Our proof also establishes the MAX SNP-hardness of the problem (again under randomized polynomial time reductions). This means that obtaining even an approximate solution to the protein folding problem, to within some fixed constant, is NP-hard. Our results are based on a general technique for replacing unbounded alphabets by finite alphabets in reductions for string folding problems. This technique has two novel aspects. The first is the essential use of the approximation hardness of the source problem in the reduction, even for the proof of NP-hardness. The second is the concept of spatial codes, a variant of classical error-correcting codes in which different codewords are required to have large `distance' from one another even when they are arbitrarily embedded in three-dimensional space.