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:(V,\ mu) 固定限量空间 $( V,\ mu) 美元, comlient $0 美元, comlient $( comlient $), 表示为$0\ box{ - Ext} [\ mu] 美元, 相当于以下优化问题: 最小化表格的函数 $( min\ limits) xxxxx 美元( xxx) xxx 美元。 最小化表格的功能 =i( x_ i) x f_ (x) i( i) +\ sum *} c_ (x_ i) 美元 (x) i) {i} {i} 立方( i) [\ max] 的计算复杂性, 以 comlifilental $( molix) comlistal max comlistal max) max max max max max max max max max max max max mox mox max mox mox mox max max max mox mox mox mox mox mox mox mox mox mox mox mox max mox mox max max mox max