arXiv 1702.044322017-02-15math.COcs.CCcs.DM

团的张量幂幂的顶点等周性和独立集稳定性

Vertex isoperimetry and independent set stability for tensor powers of cliques

Joshua Brakensiek

备注：: 24 pages, 6 figures

摘要：

$t$顶点上的团的张量幂（用$K_t^n$表示）是顶点集$\{1, ..., t\}^n$上的图，使得两个顶点$x, y \in \{1, ..., t\}^n$连通当且仅当对所有$i \in \{1, ..., n\}$都有$x_i \neq y_i$。设$K_t^n$的子集$S$的密度为$μ(S) := \frac{|S|}{t^n}$，集合$S$的顶点边界为与$S$的某个顶点关联的顶点，可能包括$S$的点.我们研究两个类似的问题，这样的图。首先，我们研究了顶点等周问题。给定密度$ν\in [0, 1]$，密度为$ν$的$K_t^n$子集的顶点边界的最小可能密度是多少？令$Φ_t(ν)$为这些最小密度的下确界，作为$n \to \infty$。我们发现一个递归关系，允许一个计算$Φ_t(ν)$在时间多项式所需的位数的精度。其次，我们研究了给定密度为$μ(I) = \frac{1}{t}(1-ε)$的独立集$I \subseteq K_t^n$，它与密度为$\frac{1}{t}$的最大独立集$J$的接近程度。我们表明，这种偏差（由$μ(I \setminus J)$测量）最多为$4ε^{\frac{\log t}{\log t - \log(t-1)}}$，只要$ε< 1 - \frac{3}{t} + \frac{2}{t^2}$。这大大改进了Alon，Dinur，Friedgut和Sudakov（2004）以及Ghandehari和Hatami（2008）的结果，后者具有$O(ε)$上限。我们还表明，指数$\frac{\log t}{\log t - \log(t-1)}$是最佳的假设$n$趋于无穷大和$ε$趋于$0$。这些方法与Ellis，Keller和Lifshitz（2016）在Kneser图和其他设置的背景下的最新工作相似。作者希望这些结果在逼近的困难性，特别是在图的近似着色和独立集问题中有潜在的应用。

英文摘要：

The tensor power of the clique on $t$ vertices (denoted by $K_t^n$) is the graph on vertex set $\{1, ..., t\}^n$ such that two vertices $x, y \in \{1, ..., t\}^n$ are connected if and only if $x_i \neq y_i$ for all $i \in \{1, ..., n\}$. Let the density of a subset $S$ of $K_t^n$ to be $μ(S) := \frac{|S|}{t^n}$, and let the vertex boundary of a set $S$ to be vertices which are incident to some vertex of $S$, perhaps including points of $S$. We investigate two similar problems on such graphs. First, we study the vertex isoperimetry problem. Given a density $ν\in [0, 1]$ what is the smallest possible density of the vertex boundary of a subset of $K_t^n$ of density $ν$? Let $Φ_t(ν)$ be the infimum of these minimum densities as $n \to \infty$. We find a recursive relation allows one to compute $Φ_t(ν)$ in time polynomial to the number of desired bits of precision. Second, we study given an independent set $I \subseteq K_t^n$ of density $μ(I) = \frac{1}{t}(1-ε)$, how close it is to a maximum-sized independent set $J$ of density $\frac{1}{t}$. We show that this deviation (measured by $μ(I \setminus J)$) is at most $4ε^{\frac{\log t}{\log t - \log(t-1)}}$ as long as $ε< 1 - \frac{3}{t} + \frac{2}{t^2}$. This substantially improves on results of Alon, Dinur, Friedgut, and Sudakov (2004) and Ghandehari and Hatami (2008) which had an $O(ε)$ upper bound. We also show the exponent $\frac{\log t}{\log t - \log(t-1)}$ is optimal assuming $n$ tending to infinity and $ε$ tending to $0$. The methods have similarity to recent work by Ellis, Keller, and Lifshitz (2016) in the context of Kneser graphs and other settings. The author hopes that these results have potential applications in hardness of approximation, particularly in approximate graph coloring and independent set problems.

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