论文标题
使用频谱剪切播放器的扩展器分解，群间边缘更少
Expander Decomposition with Fewer Inter-Cluster Edges Using a Spectral Cut Player
论文作者
论文摘要
a $（ϕ，ε）$ - 图$ g $（带有$ n $ dertices和$ m $ edges）的膨胀分解是$ v $ $ v $ in clusters $ v_1，\ ldots，v_k $，v_k $，带电导$φ（g [v_i]）\ ge ϕ $，这样的大多数是$ $ $ $ $ ^ $ ^ clusterclustersed。这种分解在许多图算法中起着至关重要的作用。我们给出了一个随机$ \ tilde {o}（m/ϕ）$ time算法用于计算$（ϕ，ϕ，ϕ \ log^2 {n}）$ - 膨胀者分解。这对$（ϕ，ϕ，ϕ \ log^3 {n}）$ - 在$ \ tilde {o}（m/ϕ）$ time中也获得了[Saranurak和Wang，Soda 2019]（SW）的时间，并带来了Optimal的Goolegarithmic因子之间的集群间边缘的数量。 SW的算法的一个关键组成部分是[Khandekar，Rao，Vazirani，JACM 2009]（KRV）的剪切游戏的不间断版本：当剪切播放器从匹配的玩家遇到不利的稀疏剪切，但仍在大面的剪裁作用时，剪切播放器并不会停止。我们改进的症结在于[Orecchia，Schulman，Vazirani，Vishnoi，Stoc 2008]的不间断版本的不间断版本（OSVV）。 OSSV的剪裁播放器使用更复杂的随机步行，微妙的潜在功能和光谱参数。 SW提出了一个明确的开放问题，设计和分析此游戏的不间断版本。
A $(ϕ,ε)$-expander-decomposition of a graph $G$ (with $n$ vertices and $m$ edges) is a partition of $V$ into clusters $V_1,\ldots,V_k$ with conductance $Φ(G[V_i]) \ge ϕ$, such that there are at most $εm$ inter-cluster edges. Such a decomposition plays a crucial role in many graph algorithms. We give a randomized $\tilde{O}(m/ϕ)$ time algorithm for computing a $(ϕ, ϕ\log^2 {n})$-expander decomposition. This improves upon the $(ϕ, ϕ\log^3 {n})$-expander decomposition also obtained in $\tilde{O}(m/ϕ)$ time by [Saranurak and Wang, SODA 2019] (SW) and brings the number of inter-cluster edges within logarithmic factor of optimal. One crucial component of SW's algorithm is non-stop version of the cut-matching game of [Khandekar, Rao, Vazirani, JACM 2009] (KRV): The cut player does not stop when it gets from the matching player an unbalanced sparse cut, but continues to play on a trimmed part of the large side. The crux of our improvement is the design of a non-stop version of the cleverer cut player of [Orecchia, Schulman, Vazirani, Vishnoi, STOC 2008] (OSVV). The cut player of OSSV uses a more sophisticated random walk, a subtle potential function, and spectral arguments. Designing and analysing a non-stop version of this game was an explicit open question asked by SW.
