2023 Fiscal Year Research-status Report
Towards a theory of smoothed analysis for distributed computing
| Project/Area Number |
21K17703
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| Research Institution | Japan Advanced Institute of Science and Technology |
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Principal Investigator |
シュワルツマン グレゴリー 北陸先端科学技術大学院大学, 先端科学技術研究科, 准教授 (20815261)
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| Project Period (FY) |
2021-04-01 – 2025-03-31
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| Keywords | Smoothed complexity / Distributed computing / graph algorithms |
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| Outline of Annual Research Achievements |
We have achieved interesting results for the problem of local max-cut on graphs with bounded arboricity. Specifically we show that the smoothed complexity of the naive FLIP algorithm for the problem is polynomial when the graph has low arboricity (e.g., it is "sparse everywhere"). This greatly broadens our understanding of the complexity landscape of the problem, as previously smoothed polynomial running times were only known for cliques and graphs with at most logarithmic degree.
While the algorithm itself appears sequential, the problem is relevant to hopfield networks - a model of associative memory. This essentially means how fast can we restore memories stored in the network (e.g, brain). However, these networks (and the brain) are inherently distributed.
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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
We believe that the results we achieved this year are a great step forward in our understanding of the smoothed complexity of the local max cut problem and hopefield networks. However, we believe that a greater results is within reach - showing a smoothed polynomial running time for general graphs, and not only graphs with bounded arboricity. This is a long standing open question and would be a huge breakthrough in the field of smoothed analysis. We will set this as the goal of the project for the current year.
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| Strategy for Future Research Activity |
The goal for this year is to show that local-max cut has smoothed polynomial complexity for general graphs. We aim to attack this problem as follows. Our results for the bounded arboricity case are the first to use structural properties of the graph to show termination guarantees. We will try to use the results we got for the bounded arboricity case to decompose the graph into a sparse and dense part and analyze each one separately. It is known that the problem has polynomial smoothed complexity on cliques, so hopefully we could leverage that.
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| Causes of Carryover |
Due to the devastating terror attack in Israel it was difficult to conduct travel visits to and from Israel where many of my collaborators reside.
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