グラフ・ネットワーク問題を解くアルゴリズムの開発

2000 Fiscal Year Annual Research Report

• Project/Area Number: 10205213
• Research Institution: Toyohashi University of Technology
• Principal Investigator: 永持 仁 豊橋技術科学大学, 工学部, 教授 (70202231)
• Co-Investigator(Kenkyū-buntansha): 石井 利昌 豊橋技術科学大学, 工学部, 助手 (30324487)
• Keywords: グラフアルゴリズム / 最小カット / 連結度 / 組合せ最適化 / 組合せ最適化ゲーム / グラフ分割 / 近似アルゴリズム / 連結度増大

Research Abstract

本研究では主に連結性に関する問題を中心にグラフ構造の解明,高速グラフアルゴリズムの開発を行った.
まず,グラフの連結性に関する構造について以下の結果を得た.重み付き無向グラフのすべての最小カットを,カクタスと呼ばれるグラフ構造に表現するアルゴリズムの計算量を軽減した.ここで使われている手法は最大隣接順序というグラフの探索法であり,すべての最小カットを計算するために最大流アルゴリズムを必要としない点が特徴である.グラフをk分割する枝集合はk-カットと呼ばれ,最小k-カットの計算は最小カット問題の拡張となる.グラフの節点数,枝数をそれぞれn,mとしたとき,グラフの2-カットを大きさの小さい順に部分的に列挙するという新しいアプローチにより,k∈{3,4,5,6}に対して従来の計算量をO(n^km)に改善した.
また,k辺連結でないグラフにおいて,カット値がk未満のカットから適当なものをいくつか互いに交叉しないように選んやると,グラフの連結度増大問題を解くために必要な情報が取り出せるという性質を示した.あらかじめこのようなカットの集合を求めておけば,連結度増大問題における解の形を細かく制御できる(たとえば,付加辺と接続する節点数を最小化できる).
近似解を求めるアルゴリズムについて以下の結果を得た.グラフの点連結度増大問題は,これまで,目標の点連結度kに対し,入力のグラフの点連結度が(k-1)である場合にしか解法が知られていなかったが,新たに,入力グラフの連結度が任意である場合のアルゴリズムを与えた.このアルゴリズムは最適解に比べ,高々2δk本しか余分に付加辺を使わない解を計算する(ここでδ=k-(入力グラフの点連結度)).さらに連結度増大問題では点連結度と辺連結度を同時に扱うものも研究し,目標値のみに依存する絶対誤差の近似アルゴリズムを得た.この他,相対誤差のアルゴリズムとして,重み付き3点連結化問題に対する(7/2)-近似,最小重み辺支配集合問題に対する2-近似,次数dのハイパーグラフ上のネットワーク設計問題に対するd・H(r_<max>)-近似のアルゴリズムを設計した(r_<max>は最大要求量,H()は調和関数).

Research Products

• [Publications] 永持仁: "初等的フローゲームの凸小生について"電子情報通信学会論文誌. J81-D-I. 666-676 (1998)
• [Publications] A.Frank: "Two-arc-disjoint paths on Eulerian digraphs"SIAM J.Discrete mathematics. 11. 557-589 (1998)
• [Publications] T.Ibaraki: "A fast algorithm for finding a maximum free multiflow in an Eulerian network and some generalization"Combinatorica. 18. 61-83 (1998)
• [Publications] H.nagamochi: "A note on minimizing submodular functions"Information Processing Letters. 67. 239-244 (1998)
• [Publications] H,Nagamochi: "Augmenting edge-connectivity over the range in o(nm) time"J.Algorithms. 30. 253-301 (1999)
• [Publications] H,Nagamochi: "An approximation of the minimum vertex cover in a graph"J.Japan Society for Industrial and Applied Mathematics. 16. 369-375 (1999)
• [Publications] H,Nagamochi: "A fast algorithm for cactus representations of minimum cuts"J.of Japan Society for Industrial and Applied Mathematics. 17. 245-264 (2000)
• [Publications] H,Nagamochi: "A faster algorithm for computing minimum 5-way and 6-way cuts in graphs"J.Combinatorial Optimization. 4. 151-169 (2000)
• [Publications] H,Nagamochi: "A7/3-approximation for the minimum weight 3-connected spanning subgraph problem"Inst.Electron,Inform,Comm.Eng.Trans.Fundamentals. E83-A. 687-691 (2000)
• [Publications] H,Nagamochi: "A fast algorithm for computing minimum 3-way and 4-way cuts"Mathematical Programming. 88. 507-520 (2000)
• [Publications] X,Deng: "Totally balanced combinatorial optimization games"Mathematical Programming,Series A. 87. 441-452 (2000)
• [Publications] H,Nagamochi: "Polyhedral structure of submodular and posi-modular systems"Discrete Applied Mathematics. 107. 165-189 (2000)
• [Publications] X.Deng: "Algorithmic aspects of the ure of combinatorial optimization games"Math.of Operations Research. 24. 751-766 (1999)
• [Publications] H,Nagamochi: "A simple prout of aminimum cut algorithm and its applications"Inst.Electron.Inform,Comm.Eng.Trans.Fund. E82-A. 2231-2236 (1999)
• [Publications] P,Eades: "Drawing clustered graphs on an orthogonal grid"J.Graph Algorithms and Application. 3. 3-29 (1999)
• [Publications] H.Nagamochi: "A simplified O(nm) time eoye-splitting algorithm in undirected graphs"Algorithmica. 26. 56-67 (2000)
• [Publications] H,Nagamochi: "Recent development of graph connectivity angmentation algorithms"Inst.Electron.Inform.Comm..Eng.Trans,Fundamentals.. E83-D. 372-383 (2000)
• [Publications] T.Ishii: "Optimal angmentation of a2-rertex-connected multigraph to a k-edge-connected and 3-rertex-connected multigraph"J.Combinatorial Optimization. 4. 35-78 (2000)
• [Publications] T.Ishii: "k-edge and 3-rertex connectivity angmentation in an arbitrary multigraph."Lecture Notes in Computer Science. 1533. 159-168 (1998)
• [Publications] H,Nagamochi: "Polyhedral structure of submodular and posi-modular systems"Lecture Notes in Computer Science. 1533. 169-178 (1998)
• [Publications] H,Nagamochi: "An efficient NC algorithm for asparse k-edge-connectivity certificate"Lecture Notes in Computer Science. 1533. 447-456 (1998)
• [Publications] H,Nagamochi: "A fast algor thin that for computing minimum 3-way and 4-way cuts"Lecture Notes in Computer Science. 1610. 377-390 (1999)
• [Publications] T.Ishii: "Augmenting an (l-1)-vertex-connected multigraph to a k-edge-connected and l-vertex-connected multigraph"Lecture Notes in Computer Science. 1643. 414-425 (1999)
• [Publications] H,Nagamochi: "An approximation for finding a smallest 2-edge-connected subgraph containing a specified spanning tree"Lecture Notes in Computer Science. 1627. 31-40 (1999)
• [Publications] H,Nagamochi: "A faster algorithm for computing minimum 5-way and 6-way cuts in graphs"Lecture Notes in Computer Science. 1627. 164-173 (1999)
• [Publications] H,Nagamochi: "Bisecting two suboets in 3-connected graphs"Lecture Notes in Computer Science. 1741. 425-434 (1999)
• [Publications] L.Zhao: "Approximating the minimum k-way cut in a graph via minimum 3-way cuts"Lecture Notes in Computer Science. 1741. 373-382 (1999)
• [Publications] T.Ishii: "On the minimum angmentation of an l-connected graph to a k-connected graph"Lecture Notes in Computer Science. 1851. 286-299 (2000)
• [Publications] T.Ishii: "Simultaneous angmentation of two graphs to an l-edge-connected graph and a biconnected graph. and a biconnected graphs"Lecture Notes in Computer Science. 1969. 326-337 (2000)
• [Publications] L.Zhao: "A primal-dual approximation algorithm for the survivable network design problem in hyper graphs"Lecture Notes in Computer Science. 2010. 478-489 (2001)
• [Publications] H,Nagamochi: "solving the single-vehicle scheduling problem for all home locations under depth-first rouling on a tree"Inst,Electron.Inform.Comm.Eng.Trans,Fundamentals. (掲載決定).
• [Publications] T,Fujito: "A 2-approximation algorithm for the minimum weightedge dominating set problem"Discrete Applied Mathematics. (掲載決定).
• [Publications] H,Nagamochi: "Bisecting two subsets in 3-connected graphs"Combinatorica. (掲載決定).
• [Publications] H,Nagamochi: "Minimum cost source location problem with vertex-connectivity requirements in digraphs"Information Processing Letters. (掲載決定).
• [Publications] T.Hasunama: "Independent spanning trees with small depths in terated line digraphs"Discrete Applied Mathematics. (掲載決定).
• [Publications] L,Zhao: "Approximating the minimum k-way cut in a graph via minimum 3-way cuts"J.Combinatorial Optimization. (掲載決定).
• [Publications] H,Nagamochi: "Angmenting a submodular and posi-modular set function by a multigraph"J.Combinatorial Optimization. (掲載決定).
• [Publications] H,Nagamochi: "An efficient NC algorithm for a sparse k-edge connectivity certificate"J.Algorithm. (掲載決定).
• [Publications] T.Ishii: "Multigraph angmentation under biconnectivity and general edge-connectivity requirements"Networks. (掲載決定).
• [Publications] H,Nagamochi: "Graph connectivity and its angmentation : application of MA orderings"Discrete Applied Mathematics. (掲載決定).
• [Publications] 永持仁: "離散構造とアルゴリズムVI第3章"藤重悟編,近代科学者. 39 (1999)