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研究生： 廖軒裕
Liao, Hsuan-Yu
論文名稱： 有色約束生成樹問題：難解的情況
The Colored Constrained Spanning Tree Problem: Intractable Conditions
指導教授： 韓永楷
Hon, Wing-Kai
口試委員: 蔡孟宗
Tsai, Meng-Tsung
王弘倫
Wang, Hung-Lung
學位類別： 碩士
Master
系所名稱： 電機資訊學院 - 資訊工程學系
Computer Science
論文出版年： 2023
畢業學年度： 111
語文別： 英文
論文頁數： 26
中文關鍵詞： 邊著色圖 、生成樹 、顏色限制 、NP困難性
外文關鍵詞： edge-colored graph, spanning tree, colored constraint, NP-hardness
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    • 在本論文中，我們研究「邊著色圖」上的「有色約束生成樹問題」和「出度有色約束生成樹問題」。這些問題旨在尋找一個生成樹，其中每個節點的任意特定顏色的鄰邊（或出邊），其邊數不超過一個給定的常數。我們證明了當輸入的圖為有向圖時，這兩個問題皆為NP困難。此外，即使輸入為有向無環圖時，有色約束生成樹問題仍然是NP困難的。

    In this thesis, we study the Colored Constrained Spanning Tree Problem (CCST problem) and the Colored Out-Constrained Spanning Tree Problem (COCST problem) on edge-colored graphs.
    These problems aim to find a spanning tree such that for each vertex, the number of incident edges (or outgoing edges) sharing any specific color is bounded by a constant.
    We demonstrate the NP-hardness of both problems when the input graphs are directed graphs.
    Additionally, even when considering directed acyclic graphs (DAGs) as input, the CCST problem remains NP-hard.

    Abstract (Chinese) i Abstract ii Acknowledgment iii Contents v 1 Introduction 1 1.1 Thesis Organization . . . . . . . . . . . . . . . 2 1.2 Related Work . . . . . . . . . . . . . . . . . . . 2 1.3 Definitions . . . . . . . . . . . . . . . . . . . 3 1.4 Our Contributions . . . . . . . . . . . . . . . . 4 2 The κ-CCST Problem on DAGs 6 2.1 Proof of Claim 1 . . . . . . . . . . . . . . . . . 11 2.2 Proof of Claim 2 . . . . . . . . . . . . . . . . . 12 2.3 Extending to Bipartite Graphs . . . . . . . . . . 14 3 The κ-COCST Problem on Directed Graphs 15 3.1 Reducing κ-CCST to κ-COCST . . . . . . . . . . . 16 3.2 Proof of Claim 3 . . . . . . . . . . . . . . . . . 18 3.3 Proof of Claim 4 . . . . . . . . . . . . . . . . . 20 4 Conclusion 23

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