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| 講演名 | 2013-05-18 11:25 Testing Subdivision-Freeness -- Property Testing Meets Structural Graph Theory -- Ken-ichi Kawarabayashi（NII）・○Yuichi Yoshida（NII/PFI）  COMP2013-15 |
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| 抄録 | （和） | Testing a property P of graphs in the bounded-degree model deals with the following problem: given a graph G of bounded degree d, we should distinguish (with probability 2/3, say) between the case that G satisfies P and the case that one should add/remove at least εdn edges of G to make it satisfy P. In sharp contrast to property testing of dense graphs, which is relatively well understood, only few properties are known to be testable with a constant number of queries in the bounded-degree model. In particular, no global monotone (i.e, closed under edge deletions) property that expander graphs can satisfy has been shown to be testable in constant time so far. In this paper, we identify for the first time a natural family of global monotone property that expander graphs can satisfy and can be efficiently tested in the bounded degree model. Specifically, we show that, for any integer t ≥ 1, K_t-subdivision-freeness is testable with a constant number of queries in the bounded-degree model. This property was not previously known to be testable even with o(n) queries. Note that an expander graph with all degree less than t - 1 does not have a K_t-subdivision. The proof is based on a novel combination of some results that develop the framework of partitioning oracles, together with structural graph theory results that develop the seminal graph minor theory by Robertson and Seymour. As far as we aware, this is the first result that bridges property testing and structural graph theory. Although we know a rough structure for graphs without H-minors from the famous graph minor theory by Robertson and Seymour, there is no corresponding structure theorem for graphs without H-subdivisions so far, even K_5-subdivision-free graphs. Therefore, subdivisions and minors are very different in a graph structural sense. |
| （英） | Testing a property P of graphs in the bounded-degree model deals with the following problem: given a graph G of bounded degree d, we should distinguish (with probability 2/3, say) between the case that G satisfies P and the case that one should add/remove at least εdn edges of G to make it satisfy P. In sharp contrast to property testing of dense graphs, which is relatively well understood, only few properties are known to be testable with a constant number of queries in the bounded-degree model. In particular, no global monotone (i.e, closed under edge deletions) property that expander graphs can satisfy has been shown to be testable in constant time so far. In this paper, we identify for the first time a natural family of global monotone property that expander graphs can satisfy and can be efficiently tested in the bounded degree model. Specifically, we show that, for any integer t ≥ 1, K_t-subdivision-freeness is testable with a constant number of queries in the bounded-degree model. This property was not previously known to be testable even with o(n) queries. Note that an expander graph with all degree less than t - 1 does not have a K_t-subdivision. The proof is based on a novel combination of some results that develop the framework of partitioning oracles, together with structural graph theory results that develop the seminal graph minor theory by Robertson and Seymour. As far as we aware, this is the first result that bridges property testing and structural graph theory. Although we know a rough structure for graphs without H-minors from the famous graph minor theory by Robertson and Seymour, there is no corresponding structure theorem for graphs without H-subdivisions so far, even K_5-subdivision-free graphs. Therefore, subdivisions and minors are very different in a graph structural sense. |
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| キーワード | （和） | property testing / structural graph theory / bounded-degree model / subdivision-freeness / / / / |
| （英） | property testing / structural graph theory / bounded-degree model / subdivision-freeness / / / / | |
| 文献情報 | 信学技報, vol. 113, no. 50, COMP2013-15, pp. 117-121, 2013年5月. | |
| 資料番号 | COMP2013-15 | |
| 発行日 | 2013-05-10 (COMP) | |
| ISSN | Print edition: ISSN 0913-5685 Online edition: ISSN 2432-6380 | |
| 著作権に ついて |
技術研究報告に掲載された論文の著作権は電子情報通信学会に帰属します．(許諾番号：10GA0019/12GB0052/13GB0056/17GB0034/18GB0034) | |
| PDFダウンロード | COMP2013-15 |
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| 研究会情報 | |
| 研究会 | COMP IPSJ-AL |
| 開催期間 | 2013-05-17 - 2013-05-18 |
| 開催地（和） | 小樽商科大学 |
| 開催地（英） | Otaru University of Commerce |
| テーマ（和） | |
| テーマ（英） | |
| 講演論文情報の詳細 | |
| 申込み研究会 | COMP |
| 会議コード | 2013-05-COMP-AL |
| 本文の言語 | 英語 |
| タイトル（和） | |
| サブタイトル（和） | |
| タイトル（英） | Testing Subdivision-Freeness |
| サブタイトル（英） | Property Testing Meets Structural Graph Theory |
| キーワード(1)（和/英） | property testing / property testing |
| キーワード(2)（和/英） | structural graph theory / structural graph theory |
| キーワード(3)（和/英） | bounded-degree model / bounded-degree model |
| キーワード(4)（和/英） | subdivision-freeness / subdivision-freeness |
| キーワード(5)（和/英） | / |
| キーワード(6)（和/英） | / |
| キーワード(7)（和/英） | / |
| キーワード(8)（和/英） | / |
| 第1著者 氏名（和/英/ヨミ） | 河原林 健一 / Ken-ichi Kawarabayashi / |
| 第1著者 所属（和/英） | 国立情報学研究所 (略称： NII) National Institute of Informatics (略称： NII) |
| 第2著者 氏名（和/英/ヨミ） | 吉田 悠一 / Yuichi Yoshida / |
| 第2著者 所属（和/英） | 国立情報学研究所/プリファードインフラストラクチャー (略称： NII/PFI) National Institute of Informatics/Preferred Infrastructure, Inc. (略称： NII/PFI) |
| 講演者 | 第2著者 |
| 発表日時 | 2013-05-18 11:25:00 |
| 発表時間 | 25分 |
| 申込先研究会 | COMP |
| 資料番号 | COMP2013-15 |
| 巻番号（vol） | vol.113 |
| 号番号（no） | no.50 |
| ページ範囲 | pp.117-121 |
| ページ数 | 5 |
| 発行日 | 2013-05-10 (COMP) |
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