2018 Fiscal Year Annual Research Report
TSP in Combinatorial Optimization and CSP in Theoretical Computer Science
| Project/Area Number |
18F18746
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| Research Institution | National Institute of Informatics |
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Principal Investigator |
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)
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| Co-Investigator(Kenkyū-buntansha) |
FULLA PETER 国立情報学研究所, ビッグデータ数理国際研究センター, 外国人特別研究員
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| Project Period (FY) |
2018-11-09 – 2021-03-31
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| Keywords | graph / matroid / CSP |
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| Outline of Annual Research Achievements |
Valuated delta-matroids (VDM) are combinatorial structures generalizing better-known 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 constant-size 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.
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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 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.
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| Strategy for Future Research Activity |
The 3-colouring problem is classic member of the NP-complete class. We are considering an approximation variant in which the given graph is guaranteed to be 3-colourable 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
state-of-the-art 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.
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Research Products
(1 results)
