In this paper, we consider the protein structure alignment problem, which is a very important problem in molecular biology Since an outline of protein structure is represented by a sequence of points in three-dimensional space, this problem is defined as the following geometric pattern matching problem: given two point sequences P and Q in three-dimensions and a real number delta > 0, find a maximum-cardinality set of point pairs such that the distance between each pair is at most delta under the condition that any translation and rotation can be applied to P. Since it is very difficult to solve this problem exactly, we consider algorithms that solve it approximately. We propose three algorithms: BASICALIGN, RANDALIGN and FRAGALIGN whose worst case time complexities are O(n(8)), O((n(7)/k(3))polylog(n)) and O(n(4)) respectively, where n denotes the size of larger input structure and k denotes the minimum size of the alignment to be obtained. All of these have the following common framework: a series of initial superpositions are computed; for each of such superpositions, a rough alignment is first computed using a dynamic programming technique, and then it is refined through an iterative improvement procedure which also uses dynamic programming; the best alignment among them is selected as an output. The difference among three algorithms lies in the methods of finding initial superpositions. BASICALIGN, RANDALIGN and FRAGALIGN use exhaustive search, random sampling technique and fragment-based search, respectively. We prove guaranteed approximation ratios (in the sense of distances between point pairs) for theoretical versions of BASICALIGN and RANDALIGN. Practical versions of RANDALIGN and FRAGALIGN were implemented and compared with a previous algorithm using real protein structure data. The experimental results show that FRIIGALIGN is best among them and it outputs good alignments quickly.