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 $${\mathcal {NP}}$$ NP -hard and $${\mathcal {APX}}$$ 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.

中文翻译：

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二维最大公共子结构问题的广义定义

两个一维序列的相似性通常由最长公共子序列（LCS）算法来衡量。然而，这些算法不能直接扩展到解决二维或更高维数据。因此，对于二维数据，使用类似 LCS 的方法计算相似性仍然值得研究。在本文中，我们参照传统的 LCS 概念，系统地给出了二维最大公共子结构 (TLCS) 问题的广义定义。通过各种匹配规则，可以定义八种可能的 TLCS 问题版本。然而，其中只有四个被证明是有效的。我们证明这四个 TLCS 问题都是 $${\mathcal {NP}}$$ NP -hard 和 $${\mathcal {APX}}$$ APX -hard。为了完成证明，