TSP in Combinatorial Optimization and CSP in Theoretical Computer Science
Project/Area Number 
18F18746

Research Category 
GrantinAid for JSPS Fellows

Allocation Type  Singleyear Grants 
Section  外国 
Research Field 
Theory of informatics

Research Institution  National Institute of Informatics 
Host Researcher 
河原林 健一 国立情報学研究所, 情報学プリンシプル研究系, 教授 (40361159)

Foreign Research Fellow 
FULLA PETER 国立情報学研究所, 情報学プリンシプル研究系, 外国人特別研究員

Project Period (FY) 
20181109 – 20210331

Project Status 
Granted (Fiscal Year 2020)

Budget Amount *help 
¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 2020: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2019: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2018: ¥400,000 (Direct Cost: ¥400,000)

Keywords  graph / matroid / CSP 
Outline of Annual Research Achievements 
Valuated deltamatroids (VDM) are combinatorial structures generalizing betterknown weighted matroids. The parity problem asks for a basis of a VDM with the minimum value subject to additional parity requirements. If the VDM is given by a direct sum of constantsize VDM, the complexity of the parity problem remains open. The special case when all the bases are assigned the same value was shown to be tractable in polynomial time by Kazda, Kolmogorov, and Rolinek (KKR) in a SODA'17 paper. We extended the KKR algorithm to allow for arbitrary values of bases. Our algorithm also runs in polynomial time, but a number of assumptions on structural properties of VDM is required in order to establish its correctness.

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 extended the KKR algorithm, as above, to allow for arbitrary values of bases. Our algorithm also runs in polynomial time, but a number of assumptions on structural properties of VDM is required in order to establish its correctness. We proved the validity of these assumptions individually; it remains to show that they also hold all combined.
Therefore, we are close to fnish the main thing in this area. We plan to consolidate the established structural properties of VDM in order to complete the proof of correctness of the candidate algorithm.

Strategy for Future Research Activity 
The 3colouring problem is classic member of the NPcomplete class. We are considering an approximation variant in which the given graph is guaranteed to be 3colourable and the task is to colour it in polynomial time using "few" colours. Previous approaches to this problem can be broadly divided into combinatorial algorithms and those employing semidefinite programming (SDP). The stateoftheart combinatorial algorithm by Kawarabayashi and Thorup uses asymptotically n^(4/11) colours (where n is the number of vertices). Our goal is to improve on this bound. We explored potential improvements in the individual components of the current algorithm. If we can show that all the affected assumptions remain valid, this will yield a decrease in the number of used colours.

Report
(1 results)
Research Products
(1 results)