This paper proposes a parameterized polynomial time approximation scheme (PTAS) for aligning two protein structures, in the case where one protein structure is represented by a contact map graph and the other by a contact map graph or a distance matrix. If the sequential order of alignment is not required, the time complexity is polynomial in the protein size and exponential with respect to two parameters D
u
/D
l
and D
c
/D
l
, which usually can be treated as constants. In particular, D
u
is the distance threshold determining if two residues are in contact or not, D
c
is the maximally allowed distance between two matched residues after two proteins are superimposed, and D
l
is the minimum inter-residue distance in a typical protein. This result clearly demonstrates that the computational hardness of the contact map based protein structure alignment problem is related not to protein size but to several parameters modeling the problem. The result is achieved by decomposing the protein structure using tree decomposition and discretizing the rigid-body transformation space. Preliminary experimental results indicate that on a Linux PC, it takes from ten minutes to one hour to align two proteins with approximately 100 residues.
