[论文解读] Decomposing a graph into expanding subgraphs

Guy Moshkovitz, A. Shapira|arXiv (Cornell University)|Jan 4, 2015

Limits and Structures in Graph Theory参考文献 31被引用 2

一句话总结

本文证明，若干著名的图分解结果——尤其是与扩展器分解和顶点分离器相关的结果——在定量界方面本质上是最优的。通过构建超立方体的随机子图，并利用度量嵌入技术进行分析，作者表明，即使在大量删边或放宽约束的条件下，也无法在当前结果的基础上进一步改善扩展性保证或边/顶点分离器界。

ABSTRACT

A paradigm that was successfully applied in the study of both pure and algorithmic problems in graph theory can be colloquially summarized as stating that any graph is close to being the disjoint union of expanders. Our goal in this paper is to show that in several of the instantiations of the above approach, the quantitative bounds that were obtained are essentially best possible. Two examples of our results are the following:• Motivated by the Unique Games Conjecture, Trevisan [FOCS '05] and Arora, Barak and Steurer [FOCS '10] showed that given a graph G, one can remove only 1% of G's edges and thus obtain a graph in which each connected component has good expansion properties. We show that in both of these decomposition results, the expansion properties they guarantee are (essentially) best possible even when one is allowed to remove 99% of G's edges. In particular, our results imply that the eigenspace enumeration approach of Arora-Barak-Steurer cannot give (even quasi-) polynomial time algorithms for unique games.• A classical result of Lipton, Rose and Tarjan from 1979 states that if F is a hereditary family of graphs and every graph in F has a vertex separator of size n/(log n)1+o(1), then every graph in F has O(n) edges. We construct a hereditary family of graphs with vertex separators of size n/(log n)1-o(1) such that not all graphs in the family have O(n) edges.The above results are obtained as corollaries of a new family of graphs, which we construct by picking random subgraphs of the hypercube, and analyze using (simple) arguments from the theory of metric embedding.

研究动机与目标

• 确定现有的扩展器分解结果在定量上是否最优。
• 研究在删去边数方面所能达到的极限，以在连通分量中实现良好扩展性。
• 检验在遗传图族中顶点分离器界是否紧致。
• 评估是否可能在当前界限之外改进唯一游戏问题的特征子空间枚举方法。
• 构造一个遗传图族，其边密度次优，尽管分离器较小，从而挑战经典界限。

提出的方法

• 构建超立方体的随机子图，作为极值例子。
• 使用度量嵌入理论中的工具分析畸变和扩展性质。
• 应用概率方法建立扩展性和分离器大小的下界。
• 分析所构造图的谱性质，以评估其扩展行为。
• 证明即使删去高达99%的边，某些图中的扩展性也无法显著改善。
• 利用超立方体的结构，推导出具有受控分离器大小但边数超线性的图族。

实验结果

研究问题

• RQ1 在允许删去99%边的前提下，扩展器分解结果中保证的扩展性能否显著优于1%的边删去？
• RQ2 唯一游戏问题的特征子空间枚举方法是否能够实现超越当前界限的多项式时间算法？
• RQ3 在具有小分离器的遗传族中，边密度的经典界限是否紧致？
• RQ4 一个遗传图族能否具有次线性分离器，却包含边数超线性的图？
• RQ5 在边删去约束下，图分解为扩展子图的定量极限是什么？

主要发现

• 即使允许删去高达99%的边，实现良好扩展性的1%边删去结果本质上是最优的。
• Arora-Barak-Steurer的特征子空间枚举方法所保证的扩展性无法显著改进，从而排除了唯一游戏问题存在（甚至准）多项式时间算法的可能性。
• 存在一个遗传图族，其顶点分离器大小为 n/(log n)^{1-o(1)}，但包含边数为 ω(n) 的图，这与经典预期相矛盾。
• 随机超立方体子图的构造在扩展性和分离器界方面具有极值性。
• 度量嵌入论证提供了一个简洁而强大的框架，用于推导扩展性和分离器性能的紧致定量下界。
• 结果表明，现有分解技术的根本限制源于图论本身的内在约束，而非算法效率低下。