Spatial codes and the hardness of string folding problems

We present a general technique for proving NP-hardness (under randomized polynomial time reductions) of 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 these problems (again under randomized polynomial time reductions). This means that obtaining even an approximate solution to the protein folding problem, to within some fixed constant factor, is NP-hard. Our technique involves replacing the symbols of an unbounded alphabet by codewords over a fixed alphabet, and 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.