2024 Fiscal Year Research-status Report
Extremal problems in graphs, their generalizations and applications to tensor networks and computer science
| Project/Area Number |
24K22830
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| Research Institution | The University of Tokyo |
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Principal Investigator |
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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| Project Period (FY) |
2024-07-31 – 2026-03-31
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| Keywords | Graph Theory / Differential Privacy / Tensor Networks |
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| Outline of Annual Research Achievements |
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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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
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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| Strategy for Future Research Activity |
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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| Causes of Carryover |
Due to using other funding for participation and travel in international conferences and joint research projects in FY 2024, the usage of the current grant has been lesser than expected. However, we have three submissions currently in the review process, and expect to require a lot of travel funding for presenting our works in international conferences in FY 2025.
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