2010 Fiscal Year Final Research Report
Developing the Algorithm Theory for Combinatorial Optimization based on Hybrid Approaches
| Project/Area Number |
20500009
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Fundamental theory of informatics
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| Research Institution | Toyohashi University of Technology |
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Principal Investigator |
FUJITO Toshihiro Toyohashi University of Technology, 大学院・工学研究科, 教授 (00271073)
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| Project Period (FY) |
2008 – 2010
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| Keywords | アルゴリズム理論 |
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| Research Abstract |
1. The tree cover problem is well studied and known to be approximable within a factor of 2 when given graphs are unweighted, whereas almost nothing is known about its approximability for directed graphs. This study considers the tree cover problem on directed layered graphs, and shows that, while it is Ω(log n) approximation hard, it can be approximated within O(log^<k-1>n) factor for graphs with k layers.
2. The independent set problem in graphs is such an NP-hard problem that is known to be hard even to approximate effectively in polynomial time. When graphs are restricted to be d-claw free, while the standard local search heuristic can approximate it within a factor of (d-1+ε)/2(ε>0) in the unweighted case, the best performance guarantee known for general weight instances is due to the Ω(n^d) time d/2-approximation algorithm, or the 2(d-1)/3-approximation algorithm running in polynomial time for any d. Either algorithm is based on the non-standard local search. This study shows the effectiveness of the standard local search for d-claw free instances under constrained weight distributions.
3. The multislope ski-rental problem1 is an extension of the classical ski-rental problem. We define the best possible competitive ratio as that of the best strategy for a given instance, and analyze its infimum and supremum over arbitrary instances. It is shown that for the (k+1)-slope problem, the infimum is (k+1)^k/((k+1)^k-k^k), implying that the competitive ratio can be no better than e/(e-1)≒1.58 no matter how many options the player may have. It is also shown that the supremum is 2.47 for k=2 and 2.75 for k=3.
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