| 研究課題/領域番号 |
24K22830
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| 研究種目 |
研究活動スタート支援
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| 配分区分 | 基金 |
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| 審査区分 |
0201:代数学、幾何学、解析学、応用数学およびその関連分野
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| 研究機関 | 東京大学 |
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研究代表者 |
MUKHERJEE SAYAN 東京大学, 大学院理学系研究科(理学部), 特任助教 (90998527)
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| 研究期間 (年度) |
2024-07-31 – 2026-03-31
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| 研究課題ステータス |
交付 (2024年度)
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| 配分額 *注記 |
2,860千円 (直接経費: 2,200千円、間接経費: 660千円)
2025年度: 1,430千円 (直接経費: 1,100千円、間接経費: 330千円)
2024年度: 1,430千円 (直接経費: 1,100千円、間接経費: 330千円)
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| キーワード | Graph Theory / Differential Privacy / Tensor Networks / Extremal Combinatorics / Computer Science |
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| 研究開始時の研究の概要 |
In extremal graph theory, people are interested in analyzing the structure of graphs that maximize or minimize some graph parameter under given constraints. We plan to analyze such extremal problems, and their applications to diverse areas such as optimizing tensor network contraction, social network analysis, and different areas of computer science. We will make use of tools such as algebra, probability theory and other mathematical tools, as well as techniques involving algorithms and programming.
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| 研究実績の概要 |
Extremal problems for graphs have significance not only in the advancement of the field of graph theory, but also in its applications in different areas of computer science and physics. The progress in this project can be summarized as follows:
(1) Tensor network contraction: Using the connection between tensor network contraction and low-congestion embeddings onto rooted binary trees, we have shown bounds on the memory complexity of contraction of arbitrary tensor networks in terms of the Laplacian matrix and the Normalized Laplacian matrix of the underlying graph. As a corollary, we also obtain upper bounds on treewidth of graphs in terms of the Laplacian eigenvalues.
(2) Differential privacy: We apply techniques of extremal and spectral graph theory together with probabilistic methods to the area of differential privacy.
(2a) We analyze differentially private graph clustering via the randomized response mechanism. Previous work in this area focuses on the stochastic block model, but we show that under mild well-clustering conditions, spectral clustering can give O(log n) privacy budget algorithms with randomized response. We also show that the privacy budget requirement of o(log n) is not possible, implying tightness of our results.
(2b) We show that under similar mild well-clustering conditions on the input graph, a modified power iteration method is able to achieve privacy with a budget of O(1). This is an improvement of our result in (2a).
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| 現在までの達成度 |
現在までの達成度
2: おおむね順調に進展している
理由
While tangible progress on extremal graph problems has been slow, generalized Turan problems on suspended paths and cycles are difficult questions. We will continue exploring more pathways to tackle this direction.
On the other hand, the applied part of this project has been progressing rather smoothly. The relationship among tensor networks, graph clustering under privacy and spectral and extremal graph theory is becoming more and more clear.
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| 今後の研究の推進方策 |
In the theoretical exploration of the generalized Turan problem for counting triangles against graph suspensions, we have tried several techniques involving algebraic constructions, probabilistic methods and graph removal lemmas. Since many of these problems are "degenerate Turan-type" and have multiple extremal constructions, this is a technical barrier that would require new ideas to overcome.
We will continue exploring different possible techniques such as subgraph counting, flag algebras and other computational methods for extremal problems. On the applied side, we will continue exploring more applications of extremal combinatorics to the fields of computer graphics, physics and computer science.
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