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 ${\tt 0\mbox{-}Ext}$ on orientable modular graphs.

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广义最小0-扩展问题和离散凸性

给定一个固定的有限度量空间 $(V,\mu)$，{\em 最小 $0$-扩展问题}，表示为 ${\tt 0\mbox{-}Ext}[\mu]$，等价于以下优化问题：最小化形式为 $\min\limits_{x\in V^n} \sum_i f_i(x_i) + \sum_{ij}c_{ij}\mu(x_i,x_j)$ 的函数，其中 $c_ {ij},c_{vi}$ 是非负成本，$f_i:V\rightarrow \mathbb R$ 是由 $f_i(x_i)=\sum_{v\in V}c_{vi}\mu(x_i) 给出的函数,v)$。${\tt 0\mbox{-}Ext}[\mu]$ 的计算复杂度最近由 Karzanov 和 Hirai 建立：如果度量 $\mu$ 是 {\em orientable Modular} 那么 ${\tt 0 \mbox{-}Ext}[\mu]$ 可以在多项式时间内求解，否则 ${\tt 0\mbox{-}Ext}[\mu]$ 是 NP-hard。为了证明易处理性部分，Hirai 开发了一种基于可定向模图的离散凸函数理论，概括了离散凸分析中的几个已知类别的函数，例如 $L^\natural$-convex 函数。我们考虑该问题的更一般版本，其中一元函数 $f_i(x_i)$ 可以另外具有 $c_{uv;i}\mu(x_i,\{u,v\})$ 形式的项，用于 $\ {u,v\}\in F$，其中 set $F\subseteq\binom{V}{2}$ 是固定的。我们通过在 $(\mu,F)$ 上提供一个明确的条件来扩展上面的复杂性分类，以使问题易于处理。为了证明易处理性部分，我们概括了 Hirai 的理论并定义了一个更大的离散凸函数类。它特别涵盖了另一类众所周知的函数，即整数点阵上的子模函数。最后，我们提高了 Hirai' 的复杂度