Studies on Diameter of High-Dimensional Polyhedra and Complexity of Simplex Method

• Project/Area Number: 15H06617
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Single-year Grants
• Research Field: Mathematical informatics
• Research Institution: Chuo University
• Principal Investigator: SUKEGAWA NORIYOSHI 中央大学, 理工学部, 助教 (20757710)
• Project Period (FY): 2015-08-28 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000); Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2015: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: 多面体理論 / 単体法 / 計算量 / 多面体 / 多項式性 / 直径 / 線形計画 / 上界

Outline of Final Research Achievements

In this research, we investigate the diameter of polyhedra. In particular, we improved the current best upper bound on the diameter of a polyhedron in terms of two parameters: dimension and number of facets of the given polyhedron. Finding a good upper bound on the diameter of a polyhedron is an outstanding open problem in the area of polyhedral combinatorics. We devised (i) a computer-assisted proof method for deriving numerically improved upper bounds and (ii) new proof techniques for deriving theoretically improved upper bounds especially for high-dimensional polyhedra.

Research Products

• [Journal Article] Improving bounds on the diameter of a polyhedron in high dimensions (2017)
  • Author(s): Noriyoshi Sukegawa
  • Journal Title: Discrete Mathemtics Volume: 印刷中 Issue: 9 Pages: 2134-2142
  • DOI: 10.1016/j.disc.2017.04.005
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Small degenerate simplices can be bad for simplex methods (2017)
  • Author(s): Shinji Mizuno, Noriyoshi Sukegawa, Antoine Deza
  • Journal Title: Journal of the Operations Research Society of Japan Volume: 印刷中
  • NAID: 130006182211
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Improving bounds on the diameter of a polyhedron in high dimensions (2016)
  • Author(s): Noriyoshi Sukegawa
  • Journal Title: arXiv preprint Volume: 1604.04039
  • Open Access / Acknowledgement Compliant
• [Journal Article] A simple proof of tail--polynomial bounds on the diameter of polyhedra (2016)
  • Author(s): Shinji Mizuno, Noriyoshi Sukegawa
  • Journal Title: arXiv preprint Volume: 1604.04338
  • Open Access / Acknowledgement Compliant
• [Journal Article] A refinement of Todd’s bound for the diameter of a polyhedron (2015)
  • Author(s): Noriyoshi Sukegawa and Tomonari Kitahara
  • Journal Title: Opereations Research Letters Volume: 43,5 Issue: 5 Pages: 534-536
  • DOI: 10.1016/j.orl.2015.07.001
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] A primal-simplex based Tardos' algorithm (2015)
  • Author(s): Shinji Mizuno, Noriyoshi Sukegawa, Antoine Deza
  • Journal Title: Operations Research Letters Volume: 43 Issue: 6 Pages: 625-628
  • DOI: 10.1016/j.orl.2015.10.002
  • Peer Reviewed / Int'l Joint Research
• [Presentation] Improving bounds on the diameter of a polyhedron in high dimensions (2016)
  • Author(s): Noriyoshi Sukegawa
  • Organizer: The Fifth International Conference on Continuous Optimization
  • Place of Presentation: 政策研究大学院大学
  • Year and Date: 2016-08-06
  • Int'l Joint Research
• [Presentation] Kalai-Kleitman不等式から得られる多面体の直径の上界について (2016)
  • Author(s): 鮏川 矩義
  • Organizer: 日本オペレーションズ・リサーチ学会
  • Place of Presentation: 慶應大学 矢上キャンパス
  • Year and Date: 2016-03-17