[论文解读] The Directed Grid Theorem

Ken‐ichi Kawarabayashi, Stephan Kreutzer|arXiv (Cornell University)|Nov 20, 2014

Advanced Graph Theory Research参考文献 31被引用 6

一句话总结

本文證明了有向網格定理，確認了Reed、Johnson、Robertson、Seymour與Thomas提出的猜想：每一個有向樹寬足夠大的有向圖都包含任意給定階數的有向網格作為蝴蝶minor。證明建立了一個函數f(k)，使得有向樹寬f(k)可保證存在階數為k的有向網格，將雙維性與結構理論延伸至有向圖，並在參數複雜性中開啟新的演算法應用。

ABSTRACT

The grid theorem, originally proved by Robertson and Seymour in Graph Minors V in 1986, is one of the most central results in the study of graph minors. It has found numerous applications in algorithmic graph structure theory, for instance in bidimensionality theory, and it is the basis for several other structure theorems developed in the graph minors project. In the mid-90s, Reed and Johnson, Robertson, Seymour and Thomas (see [Reed 97, Johnson, Robertson, Seymour, Thomas 01]), independently, conjectured an analogous theorem for directed graphs, i.e. the existence of a function f : N -> N such that every digraph of directed tree-width at least f(k) contains a directed grid of order k. In an unpublished manuscript from 2001, Johnson, Robertson, Seymour and Thomas give a proof of this conjecture for planar digraphs. But for over a decade, this was the most general case proved for the Reed, Johnson, Robertson, Seymour and Thomas conjecture. Only very recently, this result has been extended to all classes of digraphs excluding a fixed undirected graph as a minor (see [Kawarabayashi, Kreutzer 14]). In this paper, nearly two decades after the conjecture was made, we are finally able to confirm the Reed, Johnson, Robertson, Seymour and Thomas conjecture in full generality and to prove the directed grid theorem. As consequence of our results we are able to improve results in Reed et al. in 1996 [Reed, Robertson, Seymour, Thomas 96] (see also [Open Problem Garden]) on disjoint cycles of length at least l and in [Kawarabayashi, Kobayashi, Kreutzer 14] on quarter-integral disjoint paths. We expect many more algorithmic results to follow from the grid theorem.

研究动机与目标

• 解決有向圖理論中長期存在的猜想，即是否存在類似於無向網格定理的有向網格定理。
• 建立一個函數f(k)，使得每一個有向樹寬至少為f(k)的有向圖都包含階數為k的有向網格作為蝴蝶minor。
• 將無向圖理論中的結構與演算法工具（如雙維性與無關頂點技術）延伸至有向圖設定。
• 為參數複雜性中NP難問題在有向圖上的新演算法結果提供基礎。

提出的方法

• 發展了無關頂點技術的有向類比，利用結構分解與連結性論證，將高有向樹寬圖上的問題簡化。
• 使用強分割與偽柵欄分析路徑系統，並控制有向圖中的結構複雜度。
• 透過半整數與整數連結性構造遞迴地建立類似網格的大規模結構。
• 以蝴蝶minor包含關係作為定義與構造有向網格的關鍵關係。
• 結合無向圖minor理論的技術與新的有向圖專用論證，包括分割與路徑延續分析。
• 利用先前研究的成果（例如[KawarabayashiKK14, KawarabayashiK14]）建立核心結構機制。

实验结果

研究问题

• RQ1 是否存在一個函數f(k)，使得每一個有向樹寬至少為f(k)的有向圖都包含階數為k的有向網格作為蝴蝶minor？
• RQ2 無向圖的結構理論，特別是網格定理，能否延伸至有向圖？
• RQ3 有向樹寬參數是否足以保證以類似於無向樹寬的方式存在大型有向網格minor？
• RQ4 能否利用此網格定理將無關頂點方法等演算法技術適應於有向圖設定？

主要发现

• 本文確認了Reed、Johnson、Robertson、Seymour與Thomas的猜想，證明了存在一個函數f(k)，使得任何有向樹寬至少為f(k)的有向圖都包含階數為k的有向網格作為蝴蝶minor。
• 證明顯示有向網格定理在一般情況下成立，解決了有向圖結構理論中15年未解的開放問題。
• 有向網格定理啟發了新的演算法應用，例如在長度至少為l的不相交圈問題上取得改進結果，並可能擴展至固定參數可適定演算法。
• 該構造依賴於偽柵欄與連結系統的創新應用，以在有向圖中模擬網格結構。
• 函數f(k)透過連結與分割引理的遞迴應用隱式定義，未來工作預期會獲得多項式界。
• 此結果為將雙維性與結構理論延伸至有向圖提供了基礎工具，其角色類似於無向圖理論中無向網格定理的地位。