Everywhere-$δ$-dense graphs are defined as graphs on $n$ vertices in which every vertex has degree at least $δn$ for some constant $δ> 0$. Approximation schemes are vital for handling NP-hard optimization problems, but for many graph cut problems, existing PTAS algorithms often suffer from running times of $n^{f(1/\varepsilon)}$. In this paper, we bring PTASs down to EPTASs for several fundamental minimization problems on everywhere-$Ω(1)$-dense graphs. Specifically, we present the first Efficient Polynomial-Time Approximation Scheme (EPTAS), running in time $f(1/\varepsilon)n^{O(1)}$, for the ConstrainedMinCut problem under a global constraint on vertex weights, a problem that captures BalancedSeparator and SmallSetExpansion. Moreover, we give the first EPTASs for MinQuotientCut and ProductSparsestCut on everywhere-$δ$-dense graphs with integer-valued dense vertex weights; these problems generalize the four well-known problems UniformSparsestCut, EdgeExpansion, Conductance, and NormalizedCut. Our main technical contribution is an EPTAS for ConstrainedMinCut, based on the weak regularity lemma and sampling and estimation techniques. We then obtain EPTASs for MinQuotientCut and ProductSparsestCut via a unified reduction that invokes this algorithm as a subroutine. In contrast, previous works giving PTASs for these problems on everywhere-$δ$-dense graphs typically rely on powerful tools such as the Lasserre hierarchy or specific integer programming technique, which we avoid.

翻译：处处-δ-稠密图被定义为具有$n$个顶点且每个顶点度数至少为$δn$（其中$δ>0$为常数）的图。近似方案对于处理NP难优化问题至关重要，但对于许多图分割问题，现有的PTAS算法往往具有$n^{f(1/\varepsilon)}$的运行时间。在本文中，我们将处处-$Ω(1)$-稠密图上若干基本最小化问题的PTAS改进为EPTAS。具体而言，我们针对存在顶点权全局约束的ConstrainedMinCut问题（该问题涵盖了BalancedSeparator和SmallSetExpansion），提出了首个高效多项式时间近似方案（EPTAS），其运行时间为$f(1/\varepsilon)n^{O(1)}$。此外，我们还在具有整数值稠密顶点权的处处-$δ$-稠密图上，为MinQuotientCut和ProductSparsestCut问题给出了首个EPTAS；这些问题推广了著名的UniformSparsestCut、EdgeExpansion、Conductance和NormalizedCut四个问题。我们的主要技术贡献是基于弱正则性引理以及采样与估计技术的ConstrainedMinCut的EPTAS。随后，我们通过一个统一规约（将该算法作为子程序调用）得到了MinQuotientCut和ProductSparsestCut的EPTAS。相比之下，先前关于处处-$δ$-稠密图上这些问题的PTAS通常依赖于Lasserre层次或特定整数规划技术等强大工具，而我们避免了这些方法的依赖。