2022 Fiscal Year Final Research Report
Approximation of dissimilarity mappings by ultrametrics and their generalization: Theory and algorithms
| Project/Area Number |
18K11180
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 60020:Mathematical informatics-related
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| Research Institution | Shizuoka University |
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Principal Investigator |
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| Project Period (FY) |
2018-04-01 – 2023-03-31
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| Keywords | 組合せ最適化 / 超距離 / 木距離 / アルゴリズム / 系統樹推定問題 |
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| Outline of Final Research Achievements |
Given a dissimilarity mapping, to find an ultrametric which does not exceed and best approximates the dissimilarity matrix with respect to the Lp norm (the Lp-minimum increment ultrametric problem) is an important task in Phylogenetics. If p is infinite, then the problem can be solved by an efficient algorithm but if p is finite this problem is NP-hard. In this study, we give a local search algorithm for the Lp-minimum increment ultrametric problem for p=1,2.
Cycle-complete distances are a generalization of ultrametrics. In this study, we give a characterization of cycle-complete distances in terms of the associated set families. Moreover, we introduce a generalization of cycle-complete distances called k-connected complete distances and give an algorithms for finding approximations of a given dissimilarity mapping by a k-connected complete distance with respect to the L∞ norm.
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| Free Research Field |
組合せ最適化
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| Academic Significance and Societal Importance of the Research Achievements |
pが有限の場合のLp-最小増加超距離問題に対して分枝限定法や近似アルゴリズムが提案されているが，これらは計算時間の観点から解の品質に関して満足できるものではない．本研究で開発したこの問題に対する局所探索アルゴリズムは現実的な時間内で高品質な解を生み出すことができるため，系統学の研究に対して重要な貢献を与える．
閉路完全距離の，それに関連する集合族による特徴付けは，重複クラスタリング及び系統樹推定問題の分野において理論的な基礎を与える．また，任意の相違写像をk-連結完全距離によって近似するための効率的なアルゴリズムは，NP困難な最適化問題に対する近似アルゴリズムの可能性を示唆する．
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