Given a fixed finite metric space $(V,\mu)$, the {\em minimum $0$-extension problem}, denoted as ${\tt 0\mbox{-}Ext}[\mu]$, is equivalent to the following optimization problem: minimize function of the form $\min\limits_{x\in V^n} \sum_i f_i(x_i) + \sum_{ij}c_{ij}\mu(x_i,x_j)$ where $c_{ij},c_{vi}$ are given nonnegative costs and $f_i:V\rightarrow \mathbb R$ are functions given by $f_i(x_i)=\sum_{v\in V}c_{vi}\mu(x_i,v)$. The computational complexity of ${\tt 0\mbox{-}Ext}[\mu]$ has been recently established by Karzanov and by Hirai: if metric $\mu$ is {\em orientable modular} then ${\tt 0\mbox{-}Ext}[\mu]$ can be solved in polynomial time, otherwise ${\tt 0\mbox{-}Ext}[\mu]$ is NP-hard. To prove the tractability part, Hirai developed a theory of discrete convex functions on orientable modular graphs generalizing several known classes of functions in discrete convex analysis, such as $L^\natural$-convex functions. We consider a more general version of the problem in which unary functions $f_i(x_i)$ can additionally have terms of the form $c_{uv;i}\mu(x_i,\{u,v\})$ for $\{u,v\}\in F$, where set $F\subseteq\binom{V}{2}$ is fixed. We extend the complexity classification above by providing an explicit condition on $(\mu,F)$ for the problem to be tractable. In order to prove the tractability part, we generalize Hirai's theory and define a larger class of discrete convex functions. It covers, in particular, another well-known class of functions, namely submodular functions on an integer lattice. Finally, we improve the complexity of Hirai's algorithm for solving ${\tt 0\mbox{-}Ext}[\mu]$ on orientable modular graphs.

翻译：给定一个固定的有限度量空间$(V,\mu)$，记作${\tt 0\mbox{-}Ext}[\mu]$的**最小0-扩展问题**等价于以下优化问题：最小化形如$\min\limits_{x\in V^n} \sum_i f_i(x_i) + \sum_{ij}c_{ij}\mu(x_i,x_j)$的目标函数，其中$c_{ij},c_{vi}$为给定的非负代价，函数$f_i:V\rightarrow \mathbb R$由$f_i(x_i)=\sum_{v\in V}c_{vi}\mu(x_i,v)$定义。${\tt 0\mbox{-}Ext}[\mu]$的计算复杂度最近由Karzanov和Hirai确定：若度量$\mu$是**可定向模的**，则${\tt 0\mbox{-}Ext}[\mu]$可在多项式时间内求解，否则${\tt 0\mbox{-}Ext}[\mu]$是NP难的。为证明可解性部分，Hirai发展了可定向模图上的离散凸函数理论，推广了离散凸分析中若干已知函数类，如$L^\natural$-凸函数。我们考虑该问题的更一般版本，其中一元函数$f_i(x_i)$还可包含形如$c_{uv;i}\mu(x_i,\{u,v\})$的项，其中$\{u,v\}\in F$，且集合$F\subseteq\binom{V}{2}$固定。我们通过给出$(\mu,F)$上问题可解性的显式条件，扩展了上述复杂度分类。为证明可解性部分，我们推广了Hirai的理论并定义了一类更大的离散凸函数。它特别涵盖了另一类著名函数，即整数格上的子模函数。最后，我们改进了Hirai在可定向模图上求解${\tt 0\mbox{-}Ext}[\mu]$算法的复杂度。