Research on similarities and dissimilarities between the blocking and anti-blocking polyhedra

2022 Fiscal Year Final Research Report

• Project/Area Number: 18K03388
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12030:Basic mathematics-related
• Research Institution: Yamagata University
• Principal Investigator: Sakuma Tadashi 山形大学, 理学部, 教授 (60323458)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Keywords: 期待到達時間 / フィボナッチ数列 / Tutte多項式 / Pebble motion problem / safe set problem / Packing and Covering

Outline of Final Research Achievements

Proved a beautiful relationship between the expected arrival time of the squared graph of cycles and the Fibonacci number, generalizing the Tutte polynomial and the Pebble Motion Problem; overturned Ehard & Rautenbach's conjecture of connected safe set minimization in point weighted trees Showed that the problem belongs to FPTAS, solving one of the open problems of Tittmann et al [Eur J Combin 32, 2011]; partial solution of a conjecture of Cornuejols, Guenin and Margot; first in the world to show that the Tutte polynomial problem belongs to FPTAS. Gives, for the first time in the world, a combinatorial interpretation for the Tutte polynomial at (x, y) = (2, -1).

Free Research Field

離散数学、組合せ最適化

Academic Significance and Societal Importance of the Research Achievements

離散数学の幅広い分野において（期待到達時間、Tutte多項式、Pebble Motion Problem、Safe Set Problem、Clutter Packing and Covering Problemなど）に対し、複数の未解決問題を解決し、既存の枠組みの一般化を行った。特筆すべき結果としては、世界で初めて Tutte polynomialの(x, y) = (2, －1)における値に組合せ論的解釈を与えたことなどが挙げられる。