[论文解读] Adjacency Labeling Schemes for Small Classes

Édouard Bonnet, Julien Duron+2 more|arXiv (Cornell University)|Sep 7, 2024

Complexity and Algorithms in Graphs被引用 1

一句话总结

本文通過證明每一個遺傳性小圖類別都具有 O(log³n)-位的鄰接編碼方案，為小隱式圖猜想提供了強有力的證據。它確立了兩個關鍵結構性性質：弱稀疏小圖類別具有有界擴張（因此有界退化性），且所有遺傳性小圖類別的鄰域複雜度為 O(n log n)，這使得透過 VC-維度排序技術實現改進的編碼方案成為可能。

ABSTRACT

A graph class admits an implicit representation if, for every positive integer $n$, its $n$-vertex graphs have a $O(\log n)$-bit (adjacency) labeling scheme, i.e., their vertices can be labeled by binary strings of length $O(\log n)$ such that the presence of an edge between any pair of vertices can be deduced solely from their labels. The famous Implicit Graph Conjecture posited that every hereditary (i.e., closed under taking induced subgraphs) factorial (i.e., containing $2^{O(n \log n)}$ $n$-vertex graphs) class admits an implicit representation. The conjecture was recently refuted [Hatami and Hatami, FOCS '22], and does not even hold among monotone (i.e., closed under taking subgraphs) factorial classes [Bonnet et al., ICALP '24]. However, monotone small (i.e., containing at most $n! c^n$ many $n$-vertex graphs for some constant $c$) classes do admit implicit representations. This motivates the Small Implicit Graph Conjecture: Every hereditary small class admits an $O(\log n)$-bit labeling scheme. We provide evidence supporting the Small Implicit Graph Conjecture. First, we show that every small weakly sparse (i.e., excluding some fixed bipartite complete graph as a subgraph) class has an implicit representation. This is a consequence of the following fact of independent interest proved in the paper: Every weakly sparse small class has bounded expansion (hence, in particular, bounded degeneracy). Second, we show that every hereditary small class admits an $O(\log^3 n)$-bit labeling scheme, which provides a substantial improvement of the best-known polynomial upper bound of $n^{1-\varepsilon}$ on the size of adjacency labeling schemes for such classes. This is a consequence of another fact of independent interest proved in the paper: Every small class has neighborhood complexity $O(n \log n)$.

研究动机与目标

• 探討遺傳性小圖類別是否具有 O(log n)-位的鄰接編碼方案。
• 確定使高效編碼方案成為可能的小圖類別的結構性質。
• 建立小圖類別的鄰域複雜度界限與連續性結果。
• 提供支持小隱式圖猜想的證據。

提出的方法

• 證明每一個弱稀疏小圖類別都具有有界擴張，推廣了先前針對單調小圖類別與有界雙胞胎寬度類別的研究結果。
• 確立每一個遺傳性小圖類別的鄰域複雜度為 O(n log n)，此結果本身具有獨立興趣。
• 應用 Welzl 定理於低 VC-維度集合系統，以使頂點排序後其鄰域可表示為少數區間的並集。
• 利用所得的連續性界限，透過已知的從連續性到編碼的歸約關係，推導出 O(log³n)-位的編碼方案。
• 利用有界擴張意味著有界退化性的事實，強化小圖類別的結構性質。
• 結合鄰域複雜度與連續性界限，推導出最終的編碼方案大小。

实验结果

研究问题

• RQ1 所有遺傳性小圖類別是否如小隱式圖猜想所暗示的那樣，具有 O(log n)-位的鄰接編碼方案？
• RQ2 小圖類別的哪些結構性質使其能實現高效的編碼方案？
• RQ3 遺傳性小圖類別的鄰域複雜度是否可被限制在 O(n) 內，如所猜想的那樣？
• RQ4 弱稀疏小圖類別是否具有有界擴張？這對編碼方案有何含義？
• RQ5 低 VC-維度排序技術是否可用於為小圖類別構建次二次的編碼方案？

主要发现

• 每一個弱稀疏小圖類別都具有有界擴張，進而意味著有界退化性。
• 每一個遺傳性小圖類別的鄰域複雜度為 O(n log n)，此為已知類別的緊緻界限。
• 任何屬於遺傳性小圖類別的 n-頂點圖的連續性為 O(log²n)。
• 每一個遺傳性小圖類別都存在 O(log³n)-位的鄰接編碼方案。
• 鄰域複雜度界限對已知類別（如有界雙胞胎寬度與有界樹寬）而言是緊緻的。
• 該方法表明，若鄰域複雜度為 O(n)，則可能實現 O(log²n)-位的編碼方案，如所猜想的那樣。