Generalized minimum 0-extension problem and discrete convexity

Given a fixed finite metric space \((V,\mu )\), the minimum 0-extension problem, denoted as \(\mathtt{0\hbox {-}Ext}[{\mu }]\), is equivalent to the following optimization problem: minimize function of the form \(\min \nolimits _{x\in V^n} \sum _i f_i(x_i) + \sum _{ij} c_{ij}\hspace{0.5pt}\mu (x_i,x_j)\) where \(f_i:V\rightarrow \mathbb {R}\) are functions given by \(f_i(x_i)=\sum _{v\in V} c_{vi}\hspace{0.5pt}\mu (x_i,v)\) and \(c_{ij},c_{vi}\) are given nonnegative costs. The computational complexity of \(\mathtt{0\hbox {-}Ext}[{\mu }]\) has been recently established by Karzanov and by Hirai: if metric \(\mu \) is orientable modular then \(\mathtt{0\hbox {-}Ext}[{\mu }]\) can be solved in polynomial time, otherwise \(\mathtt{0\hbox {-}Ext}[{\mu }]\) 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 \(c_{uv;i}\hspace{0.5pt}\mu (x_i,\{u,v\})\) for \(\{u,\!\hspace{0.5pt}\hspace{0.5pt}v\}\in F\), where set \(F\subseteq \left( {\begin{array}{c}V\\ 2\end{array}}\right) \) 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 \(\mathtt{0\hbox {-}Ext}[{\mu }]\) on orientable modular graphs.