Generalized minimum 0-extension problem and discrete convexity

Abstract

AbstractGiven a fixed finite metric space $$(V,\mu )$$ ( V , μ ) , the minimum 0-extension problem, denoted as $$\mathtt{0\hbox {-}Ext}[{\mu }]$$ 0 - Ext [ μ ] , is equivalent to the following optimization problem: minimize function of the form $$\min \nolimits _{x\in V^n} \sum _i f_i(x_i) + \sum _{ij} c_{ij}\hspace{0.5pt}\mu (x_i,x_j)$$ min x ∈ V n ∑ i f i ( x i ) + ∑ ij c ij μ ( x i , x j ) where $$f_i:V\rightarrow \mathbb {R}$$ f i : V → R are functions given by $$f_i(x_i)=\sum _{v\in V} c_{vi}\hspace{0.5pt}\mu (x_i,v)$$ f i ( x i ) = ∑ v ∈ V c vi μ ( x i , v ) and $$c_{ij},c_{vi}$$ c ij , c vi are given nonnegative costs. The computational complexity of $$\mathtt{0\hbox {-}Ext}[{\mu }]$$ 0 - Ext [ μ ] has been recently established by Karzanov and by Hirai: if metric $$\mu $$ μ is orientable modular then $$\mathtt{0\hbox {-}Ext}[{\mu }]$$ 0 - Ext [ μ ] can be solved in polynomial time, otherwise $$\mathtt{0\hbox {-}Ext}[{\mu }]$$ 0 - Ext [ μ ] 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 $$L^\natural $$ L ♮ -convex functions. We consider a more general version of the problem in which unary functions $$f_i(x_i)$$ f i ( x i ) can additionally have terms of the form $$c_{uv;i}\hspace{0.5pt}\mu (x_i,\{u,v\})$$ c u v ; i μ ( x i , { u , v } ) for $$\{u,\!\hspace{0.5pt}\hspace{0.5pt}v\}\in F$$ { u , v } ∈ F , where set $$F\subseteq \left( {\begin{array}{c}V\\ 2\end{array}}\right) $$ F ⊆ V 2 is fixed. We extend the complexity classification above by providing an explicit condition on $$(\mu ,F)$$ ( μ , 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 $$\mathtt{0\hbox {-}Ext}[{\mu }]$$ 0 - Ext [ μ ] on orientable modular graphs.