Generalized minimum 0-extension problem and discrete convexity
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Given a fixed finite metric space (V,μ), the minimum 0-extension problem, denoted as 0Ext[μ], is equivalent to the following optimization problem: minimize function of the form min_x∈ V^n∑_i f_i(x_i) + ∑_ijc_ijμ(x_i,x_j) where c_ij,c_vi are given nonnegative costs and f_i:V→ℝ are functions given by f_i(x_i)=∑_v∈ Vc_viμ(x_i,v). The computational complexity of 0Ext[μ] has been recently established by Karzanov and by Hirai: if metric μ is orientable modular then 0Ext[μ] can be solved in polynomial time, otherwise 0Ext[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^♮-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;iμ(x_i,{u,v}) for {u,v}∈ F, where set F⊆V2 is fixed. We extend the complexity classification above by providing an explicit condition on (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 0Ext[μ] on orientable modular graphs.
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