In the $0$-Extension problem, we are given an edge-weighted graph $G=(V,E,c)$, a set $T\subseteq V$ of its vertices called terminals, and a semi-metric $D$ over $T$, and the goal is to find an assignment $f$ of each non-terminal vertex to a terminal, minimizing the sum, over all edges $(u,v)\in E$, the product of the edge weight $c(u,v)$ and the distance $D(f(u),f(v))$ between the terminals that $u,v$ are mapped to. Current best approximation algorithms on $0$-Extension are based on rounding a linear programming relaxation called the \emph{semi-metric LP relaxation}. The integrality gap of this LP, with best upper bound $O(\log |T|/\log\log |T|)$ and best lower bound $\Omega((\log |T|)^{2/3})$, has been shown to be closely related to the best quality of cut and flow vertex sparsifiers. We study a variant of the $0$-Extension problem where Steiner vertices are allowed. Specifically, we focus on the integrality gap of the same semi-metric LP relaxation to this new problem. Following from previous work, this new integrality gap turns out to be closely related to the quality achievable by cut/flow vertex sparsifiers with Steiner nodes, a major open problem in graph compression. Our main result is that the new integrality gap stays superconstant $\Omega(\log\log |T|)$ even if we allow a super-linear $O(|T|\log^{1-\varepsilon}|T|)$ number of Steiner nodes.

翻译：在$0$-Extension问题中，给定一个边赋权图$G=(V,E,c)$、其顶点子集$T\subseteq V$（称为终端集），以及$T$上的半度量$D$，目标是找到每个非终端顶点到某个终端的分配$f$，使得所有边$(u,v)\in E$上边权$c(u,v)$与$u,v$所映射终端间距离$D(f(u),f(v))$的乘积之和最小。当前$0$-Extension的最佳近似算法基于对称为半度量线性规划松弛的线性规划进行舍入。该线性规划的积分间隙，其上界最佳为$O(\log |T|/\log\log |T|)$，下界最佳为$\Omega((\log |T|)^{2/3})$，已被证明与割和流顶点稀疏化器的最佳质量密切相关。我们研究允许Steiner顶点的$0$-Extension问题变体。具体而言，我们聚焦于同一半度量线性规划松弛在该新问题中的积分间隙。根据先前的工作，这一新积分间隙恰好与带Steiner节点的割/流顶点稀疏化器可达质量密切相关，这是图压缩领域中的一个重大开放问题。我们的主要结果是：即使允许使用超线性数量$O(|T|\log^{1-\varepsilon}|T|)$的Steiner节点，该新积分间隙仍保持超常数$\Omega(\log\log |T|)$。