For a metric $\mu$ on a finite set $T$, the minimum 0-extension problem 0-Ext$[\mu]$ is defined as follows: Given $V\supseteq T$ and $\ c:{V \choose 2}\rightarrow \mathbf{Q_+}$, minimize $\sum c(xy)\mu(\gamma(x),\gamma(y))$ subject to $\gamma:V\rightarrow T,\ \gamma(t)=t\ (\forall t\in T)$, where the sum is taken over all unordered pairs in $V$. This problem generalizes several classical combinatorial optimization problems such as the minimum cut problem or the multiterminal cut problem. Karzanov and Hirai established a complete classification of metrics $\mu$ for which 0-Ext$[\mu]$ is polynomial time solvable or NP-hard. This result can also be viewed as a sharpening of the general dichotomy theorem for finite-valued CSPs (Thapper and \v{Z}ivn\'{y} 2016) specialized to 0-Ext$[\mu]$. In this paper, we consider a directed version $\overrightarrow{0}$-Ext$[\mu]$ of the minimum 0-extension problem, where $\mu$ and $c$ are not assumed to be symmetric. We extend the NP-hardness condition of 0-Ext$[\mu]$ to $\overrightarrow{0}$-Ext$[\mu]$: If $\mu$ cannot be represented as the shortest path metric of an orientable modular graph with an orbit-invariant ``directed'' edge-length, then $\overrightarrow{0}$-Ext$[\mu]$ is NP-hard. We also show a partial converse: If $\mu$ is a directed metric of a modular lattice with an orbit-invariant directed edge-length, then $\overrightarrow{0}$-Ext$[\mu]$ is tractable. We further provide a new NP-hardness condition characteristic of $\overrightarrow{0}$-Ext$[\mu]$, and establish a dichotomy for the case where $\mu$ is a directed metric of a star.

翻译：对于有限集$T$上的度量$\mu$，最小0-扩展问题0-Ext$[\mu]$定义如下：给定$V\supseteq T$和权重函数$ c:{V \choose 2}\rightarrow \mathbf{Q_+}$，在约束$\gamma:V\rightarrow T,\ \gamma(t)=t\ (\forall t\in T)$下最小化$\sum c(xy)\mu(\gamma(x),\gamma(y))$，其中求和遍历$V$中所有无序对。该问题推广了若干经典组合优化问题，如最小割问题或多终端割问题。Karzanov与Hirai建立了关于度量$\mu$的完整分类，刻画了使0-Ext$[\mu]$可在多项式时间内求解或为NP困难的条件。该结果亦可视为有限值CSPs一般二分定理（Thapper与\v{Z}ivn\'{y} 2016）在0-Ext$[\mu]$上的精细化。本文考虑最小0-扩展问题的有向版本$\overrightarrow{0}$-Ext$[\mu]$，其中$\mu$与$c$均无需对称性假设。我们将0-Ext$[\mu]$的NP困难条件推广至$\overrightarrow{0}$-Ext$[\mu]$：若$\mu$不能表示为某可定向模图在轨道不变的"有向"边权下的最短路径度量，则$\overrightarrow{0}$-Ext$[\mu]$是NP困难的。我们还建立了部分逆命题：若$\mu$是某模格在轨道不变有向边权下的有向度量，则$\overrightarrow{0}$-Ext$[\mu]$是可解的。我们进一步给出了$\overrightarrow{0}$-Ext$[\mu]$特有的新型NP困难条件，并针对$\mu$为星形有向度量的情形建立了二分定理。