Generalized minimum 0-extension problem and discrete convexity

Given a fixed finite metric space (V,μ ) , the minimum 0-extension problem , denoted as 0-𝙴𝚡𝚝[μ] , is equivalent to the following optimization problem: minimize function of the form min _x∈ V^n∑ _i f_i(x_i) + ∑ _ij c_ijμ (x_i,x_j) where f_i:V→ℝ are functions given by f_i(x_i)=∑ _v∈ V c_viμ (x_i,v) and c_ij,c_vi are given nonnegative costs. The computational complexity of 0-𝙴𝚡𝚝[μ] has been recently established by Karzanov and by Hirai: if metric μ is orientable modular then 0-𝙴𝚡𝚝[μ] can be solved in polynomial time, otherwise 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^♮ -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⊆( [ V; 2 ]) 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 0-𝙴𝚡𝚝[μ] on orientable modular graphs.