2004 Fiscal Year Final Research Report Summary
Enumeration and Sampling Based Algorithms for Combinatorial Optimization Problems
| Project/Area Number |
13680510
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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 |
社会システム工学
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| Research Institution | The University of Tokyo |
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Principal Investigator |
MATUURA Shiro The University of Tokyo, Graduate School of Science and Technology, Assistant Professor, 大学院・情報理工学系研究科, 助手 (00332619)
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| Co-Investigator(Kenkyū-buntansha) |
MATSUI Tomomi The University of Tokyo, Graduate School of Science and Technology, Associate Professor, 大学院・情報理工学系研究科, 助教授 (30270888)
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| Project Period (FY) |
2001 – 2004
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| Keywords | combinatorial optimization / enumeration algorithm / scheduling / approximation algorithm / sampling method |
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| Research Abstract |
The following results are obtained in this research project.
1.Integer quadratic programming problems
We proposed a 0.935-approximation algorithm for MAX 2SAT and a 0.863-approximation algorithm for MAX DICUT. The approximation ratios improve upon the recent results of Zwick, which are equal to 0.93109 and 0.8596439254 respectively. Also proposed are derandomizad versions of the same approximation ratios. We note that these approximation ratios are obtained by numerial computaiton rather than theoretical proof. The algorithms are based on the SDP relaxation proposed by Goemans and Williamson but do not use the 'rotation' technique proposed by Feign and Goemans. The improvements in the approximation ratios are obtained by the technique of 'hyperplane separation with skewed distribution function on the sphere'.
2.Sports Scheduling
The break minimization problem is to find a home-away assignment that minimizes the number of breaks for a given schedule of round-robin tournament. In a recent pa
… More
per, Elf, Junger and Rinaldi conjectured that there exists a polynomial time algorithm to determine whether the optimal value of the given break minimization problem is less than the number of teams. We proved the conjecture affirmatively by showing that the determining problem is solvable in polynomial time. Our approach is to transform the determining problem to instances of 2SAT. We also showed that the break minimization problem can be formulated as MAX RES CUT and MAX 2SAT. We applied Goemans and Williamson's approximation algorithm for MAX 2SAT based on positive semidefinite programming relaxation. Our computational experiments show that the approximation algorithm finds good solutions in practical computational time.
3.Sampling and counting problems
we proposed a new counting scheme for m x n contingency tables. Our scheme is a modification of Dyer and Greenhill's scheme. We can estimate not only the sizes of error, but also the sizes of the bias of the number of tables obtained by our scheme, on the assumption that we have an approximate sampler. For sampling 2x2x...x2xJ contingency tables, we proposed two Markov chains. Stationary distributions of our chains are the uniform distribution and a conditional multinomial distribution We showed that our chains mix rapidly. For two-rowed contingency tables, we propose a polynomial time perfect (exact) sampling algorithm. Our algorithm is a Las Vegas type randomized algorithm and the expected running time is bounded by a polynomial of input size. The main idea of our algorithm is the monotone coupling from the past (monotone CFTP) algorithm proposed by Prop and Wilson. Less
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