2004 Fiscal Year Final Research Report Summary
Development of Algorithm Theory Based on Mathematical Programming and Probability Tyeory
| Project/Area Number |
15500008
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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 (2004)
Nagoya University (2003) |
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Principal Investigator |
FUJITO Toshihiro Toyohashi University of Technology, Engineering, Professor, 工学部, 教授 (00271073)
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| Project Period (FY) |
2003 – 2004
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| Keywords | Approximation Algorithm / Submodular Set Cover / Covering Problem / Set Packing Problem |
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| Research Abstract |
・After introducing a generalization of the vertex cover problem on graphs with vertex and edge constraints, we show it to be polynomially approximable within a factor of 2,using an extended version of the submodular set cover algorithm.
・The tree cover problem is known approximable within a factor of 2 only when all the edge costs are uniform, whereas some related problems such as vertex cover and edge dominating set are 2-approximable under general costs. We develop a primal-dual algorithm for tree cover and show that its approximation factor is 2 when only two kinds of edge weights, differing by a multiplicative factor of at least 2,are allowed.
・While several 2-approximation NC algorithms are known for the vertex cover problem on graphs, no such algorithm is known for the connected vertex cover problem. We develop 2-approximation NC and RNC algorithms for tree cover and connected vertex cover.
・It is shown that the set multicover problem can be approximated within a factor of H(k)-1/6 by a modified greedy algorithm newly developed for set multicover.
・We develop an efficient and purely combinatorial algorithm for the covering 0-1 integer program problem, and show its performance is in general as good as those of the rounding algorithms.
・Extending a local search heuristic for the unweighted set packing problem, it is shown that the k-set packing problem with weights 1 and w such that w【greater than or equal】2 canbe approximated within a factor of k/2+ε.
・We introduce a production planning problem called the capacitated supply-demand (CSD) problem, and, to analyze its structural properties, extend the submodular set cover problem to the one, called submodular integer cover (SIC), with submodular constraints on integer vectors instead of on subsets. By applying the primal-dual heuristic for SIC to CSD, it is shown that CSD can be approximated by a factor dependent on the network structure but not on any numerical value.
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