For a metric $\mu$ on a finite set $T$, the minimum 0-extension problem $\mathbf{0}$-$\mathbf{Ext}\left[\mu \right]$ is defined as follows: Given $V\supseteq T$ and $c:\left(\begin{array}{c}\hfill V\hfill \\ \hfill 2\hfill \end{array}\right)\to {\mathbb{Q}}_{+}$, minimize $\sum c\left(xy\right)\mu (\gamma \left(x\right),\gamma \left(y\right))$ subject to $\gamma :V\to T,\gamma \left(t\right)=t(\forall t\in T)$, 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 $\mathbf{0}$-$\mathbf{Ext}\left[\mu \right]$ 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 Živný 2016) specialized to $\mathbf{0}$-$\mathbf{Ext}\left[\mu \right]$.

In this paper, we consider a directed version $\vec{\mathbf{0}}$-$\mathbf{Ext}\left[\mu \right]$ of the minimum 0-extension problem, where $\mu$ and $c$ are not assumed to be symmetric. We extend the NP-hardness condition of $\mathbf{0}$-$\mathbf{Ext}\left[\mu \right]$ to $\vec{\mathbf{0}}$-$\mathbf{Ext}\left[\mu \right]$: If $\mu$ cannot be represented as the shortest path metric of an orientable modular graph with an orbit-invariant “directed” edge-length, then $\vec{\mathbf{0}}$-$\mathbf{Ext}\left[\mu \right]$ 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 $\vec{\mathbf{0}}$-$\mathbf{Ext}\left[\mu \right]$ is tractable. We further provide a new NP-hardness condition characteristic of $\vec{\mathbf{0}}$-$\mathbf{Ext}\left[\mu \right]$, and establish a dichotomy for the case where $\mu$ is a directed metric of a star.

对于指标 $\mu$ 在有限集上 $Ť$，最小0扩展问题 $\mathbf{0}$--$\mathbf{分机}\left[\mu \right]$ 定义如下： $\mathrm{伏特}\supseteq Ť$ 和 $C：\left(\begin{array}{c}\hfill \mathrm{伏特}\hfill \\ \hfill \mathrm{2个}\hfill \end{array}\right)\to {\mathbb{问}}_{+}$， 最小化 $\sum C（Xÿ）\mu （\gamma （X），\gamma （ÿ））$ 服从 $\gamma ：\mathrm{伏特}\to Ť，\gamma （Ť）=Ť（\forall Ť\in Ť）$，其中的总和取于的所有无序对中 $\mathrm{伏特}$。该问题概括了几个经典的组合优化问题，例如最小割问题或多端子割问题。Karzanov和Hirai建立了指标的完整分类$\mu$ 为此 $\mathbf{0}$--$\mathbf{分机}\left[\mu \right]$是多项式时间可解的或NP难解的。该结果也可以看作是对专门用于有限值CSP的一般二分法定理的强化（Thapper andŽivný2016）。$\mathbf{0}$--$\mathbf{分机}\left[\mu \right]$。

在本文中，我们考虑定向版本 $\vec{\mathbf{0}}$--$\mathbf{分机}\left[\mu \right]$ 最小0扩展问题 $\mu$ 和 $C$不假定是对称的。我们扩展了NP硬度条件$\mathbf{0}$--$\mathbf{分机}\left[\mu \right]$ 到 $\vec{\mathbf{0}}$--$\mathbf{分机}\left[\mu \right]$： 如果 $\mu$ 不能表示为具有轨道不变的“有向”边长的可定向模块图的最短路径度量，则 $\vec{\mathbf{0}}$--$\mathbf{分机}\left[\mu \right]$是NP难的。我们还展示了部分相反的情况：$\mu$ 是具有轨道不变的有向边长的模块化晶格的有向度量，然后 $\vec{\mathbf{0}}$--$\mathbf{分机}\left[\mu \right]$很容易处理。我们进一步提供了一种新的NP硬度条件$\vec{\mathbf{0}}$--$\mathbf{分机}\left[\mu \right]$，并针对以下情况建立二分法 $\mu$ 是恒星的有向度量。