一維資料的相似度通常使用最長共同子序列演算法來計算。然而，這些演算法並不能直接使用來計算二維以上的資料。因此提出二維最大共同子結構問題來計算二維資料的相似度。在2016年，定義了8種不同版本的二維最大共同子結構問題，其中4種已經被證明為有效的配對條件，而其餘4種為無效的配對條件。此外，其中2種版本已被證明為NP難題，並且猜測另外2種版本同樣為NP難題。在本論文中，我們證明了未被證明的2種版本同樣維NP難題。接著我們證明4種有效版本的二維最大子結構問題皆為APX難題。

The similarity of one-dimensional data is usually measured by the longest common subsequence (LCS) algorithms. However, these algorithms cannot be directly extended to solve the case with two or higher dimensional data. The two-dimensional largest common substructure (TLCS) problem was therefore proposed to compute the similarity of two-dimensional data. In 2016, Chan et al. defined eight different versions of the TLCS problem, and four of them were shown to be valid for pattern matching, while the other four are invalid. In addition, Chan et al. showed that two versions of them are NP-hard, and left a conjecture that the other two are also NP-hard. In this thesis, we prove that the remaining two versions of the TLCS problem are NP-hard, showing the correctness of Chan's conjecture. Moreover, we prove that the four valid versions are all APX-hard.

[THESIS VERIFICATION FORM+i]
[THESIS AUTHORIZATION FORM+iii]
[ACKNOWLEDGMENTS+iv]
[CHINESE ABSTRACT+v]
[ABSTRACT+vi]
[LIST OF FIGURES+ix]
[LIST OF TABLES+x]
[1 Introduction+1]
[2 Preliminary+3]
[2.1 Notations+3]
[2.2 The Two-dimensional Largest Common Substructure Problem+4]
[2.3 Approximability Classes+7]
[2.4 The Maximum 3-satisfiability Bounded Problem+10]
[2.5 The Maximum 3-dimensional Matching Bounded Problem+10]
[3 Proofs of NP-hardness and APX-hardness+12]
[3.1 APX-hardness for P(ENL) and P(ENE)+12]
[3.2 NP-hardness of P(LOL) and P(LOE)+16]
[3.3 APX-hardness of P(LOL) and P(LOE)+19]
[4 Conclusion+21]
[BIBLIOGRAPHY+22]

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