The Analysis of Computational Complexity of Discrete Problems
Project/Area Number 
13640139

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  Nihon University 
Principal Investigator 
TODA Seinosuke Nihon University, Dept. Computer Science and System Analysis, Professor, 文理学部, 教授 (90172163)

CoInvestigator(Kenkyūbuntansha) 
CHEN Zhizhong Tokyo Denki University, Dept. Mathematical Science, Associate Professor, 理工学部, 助教授 (00242933)
TANI Seiichi Nihon University, Dept. Computer Science and System Analysis, Associate Professor, 文理学部, 助教授 (70266708)
YAKU Takeo Nihon University, Dept. Computer Science and System Analysis, Professor, 文理学部, 教授 (90102821)

Project Period (FY) 
2001 – 2003

Project Status 
Completed (Fiscal Year 2003)

Budget Amount *help 
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2003: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2002: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2001: ¥1,800,000 (Direct Cost: ¥1,800,000)

Keywords  Computational Complexity / graph theory / graph isomorphism / self avoiding walk / polynomial time algorithm / #P complete / chordal graph / grid graph / 2次元格子グラフ / 超立方体グラフ / selfavoiding walk / 数え上げ問題 / 結び目 / 自明性判定問題 / ブレイド / 共役問題 / 絡み目 / 同型写真像 / 多項式時間 / マッチング / 近似アルゴリズム / アルゴリズム / 多次元格子 / 超立方体 / 数え上げ / 多項式 
Research Abstract 
In this research project, we mainly investigate the computational complexity of discrete problems. In particular, we dealt with graph Isomorphism problem and the problem of counting self avoiding walks in graphs. At this point, exploring the. precise complexity of the problems has remained to be important open questions in computational complexity theory while many researches were done so far. We currently believe that investigating their computational complexity may give us a new insight on the structure of computations. In this research project, we obtained several results mentioned as follows. Related to the graph isomorphism problem, we first showed that the problem of counting graph isomorphisms among partial ktrees was computable in polynomial time with developing a dynamic programming algorithm. In this algorithm, we had to compute the permanent of bipartite graphs, which is the number of perfect matching in bipartite graphs. In usual, such a computation has appeared to be hard. But, in our case, we found that bipartite graphs concerned had a strong symmetry, and then we succeeded to design an efficient algorithm for computing the permanent. We further showed that the graph isomorphism problem on the class of chordal bipartite graphs and on the class of strongly chordal graphs remained to be GI complete.. These results refine the previous knowledge on the complexity. of the problem. We also showed that the problems of counting selfavoiding walks both in twodimensional grid graphs and in hypercube graphs were complete for #P. This is a first result concerned on the complexity of the problem. We further showed that the problem was #EXPcomplete in case that an input graph was given in a succinct representation form. We further designed a lineartime algorithm for 7coloring 1 planar graphs, and we study a possibility of developing software systems with using graph grammar theory.

Report
(4 results)
Research Products
(23 results)