The Generalized Definitions of the Two-Dimensional Largest Common Substructure Problems

The similarity of two one-dimensional sequences is usually measured by the longest common subsequence (LCS) algorithms. However, these algorithms cannot be directly extended to solve the two or higher dimensional data. Thus, for the two-dimensional data, computing the similarity with an LCS-like approach remains worthy of investigation. In this paper, we utilize a systematic way to give the generalized definition of the two-dimensional largest common substructure (TLCS) problem by referring to the traditional LCS concept. With various matching rules, eight possible versions of TLCS problems may be defined. However, only four of them are shown to be valid. We prove that all of these four TLCS problems are NP -hard and APX -hard. To accomplish the proofs, two of the TLCS problems are reduced from the 3-satisfiability problem, and the other two are reduced from the 3-dimensional matching problem.