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Program."],"published-print":{"date-parts":[[2025,1]]},"abstract":"Abstract<\/jats:title>\n Given a fixed finite metric space \n \n $$(V,\\mu )$$<\/jats:tex-math>\n \n \n (<\/mml:mo>\n V<\/mml:mi>\n ,<\/mml:mo>\n \u03bc<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, the minimum 0-extension problem<\/jats:italic>, denoted as \n \n $$\\mathtt{0\\hbox {-}Ext}[{\\mu }]$$<\/jats:tex-math>\n \n \n 0<\/mml:mn>\n -<\/mml:mtext>\n Ext<\/mml:mi>\n [<\/mml:mo>\n \u03bc<\/mml:mi>\n ]<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, is equivalent to the following optimization problem: minimize function of the form \n \n $$\\min \\nolimits _{x\\in V^n} \\sum _i f_i(x_i) + \\sum _{ij} c_{ij}\\hspace{0.5pt}\\mu (x_i,x_j)$$<\/jats:tex-math>\n \n \n \n min<\/mml:mo>\n \n x<\/mml:mi>\n \u2208<\/mml:mo>\n \n V<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:msub>\n \n \u2211<\/mml:mo>\n i<\/mml:mi>\n <\/mml:msub>\n \n f<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n \n x<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n +<\/mml:mo>\n \n \u2211<\/mml:mo>\n \n ij<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n c<\/mml:mi>\n \n ij<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n \u03bc<\/mml:mi>\n \n (<\/mml:mo>\n \n x<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n ,<\/mml:mo>\n \n x<\/mml:mi>\n j<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> where \n \n $$f_i:V\\rightarrow \\mathbb {R}$$<\/jats:tex-math>\n \n \n \n f<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n :<\/mml:mo>\n V<\/mml:mi>\n \u2192<\/mml:mo>\n R<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> are functions given by \n \n $$f_i(x_i)=\\sum _{v\\in V} c_{vi}\\hspace{0.5pt}\\mu (x_i,v)$$<\/jats:tex-math>\n \n \n \n f<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n \n x<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n \n \u2211<\/mml:mo>\n \n v<\/mml:mi>\n \u2208<\/mml:mo>\n V<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n c<\/mml:mi>\n \n vi<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n \u03bc<\/mml:mi>\n \n (<\/mml:mo>\n \n x<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n ,<\/mml:mo>\n v<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and \n \n $$c_{ij},c_{vi}$$<\/jats:tex-math>\n \n \n \n c<\/mml:mi>\n \n ij<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n ,<\/mml:mo>\n \n c<\/mml:mi>\n \n vi<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> are given nonnegative costs. The computational complexity of \n \n $$\\mathtt{0\\hbox {-}Ext}[{\\mu }]$$<\/jats:tex-math>\n \n \n 0<\/mml:mn>\n -<\/mml:mtext>\n Ext<\/mml:mi>\n [<\/mml:mo>\n \u03bc<\/mml:mi>\n ]<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> has been recently established by Karzanov and by Hirai: if metric \n \n $$\\mu $$<\/jats:tex-math>\n \n \u03bc<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is orientable modular<\/jats:italic> then \n \n $$\\mathtt{0\\hbox {-}Ext}[{\\mu }]$$<\/jats:tex-math>\n \n \n 0<\/mml:mn>\n -<\/mml:mtext>\n Ext<\/mml:mi>\n [<\/mml:mo>\n \u03bc<\/mml:mi>\n ]<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> can be solved in polynomial time, otherwise \n \n $$\\mathtt{0\\hbox {-}Ext}[{\\mu }]$$<\/jats:tex-math>\n \n \n 0<\/mml:mn>\n -<\/mml:mtext>\n Ext<\/mml:mi>\n [<\/mml:mo>\n \u03bc<\/mml:mi>\n ]<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> 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 \n \n $$L^\\natural $$<\/jats:tex-math>\n \n \n L<\/mml:mi>\n \u266e<\/mml:mo>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-convex functions. We consider a more general version of the problem in which unary functions \n \n $$f_i(x_i)$$<\/jats:tex-math>\n \n \n \n f<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n \n x<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> can additionally have terms of the form \n \n $$c_{uv;i}\\hspace{0.5pt}\\mu (x_i,\\{u,v\\})$$<\/jats:tex-math>\n \n \n \n c<\/mml:mi>\n \n u<\/mml:mi>\n v<\/mml:mi>\n \u037e<\/mml:mo>\n i<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n \u03bc<\/mml:mi>\n \n (<\/mml:mo>\n \n x<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n ,<\/mml:mo>\n \n {<\/mml:mo>\n u<\/mml:mi>\n ,<\/mml:mo>\n v<\/mml:mi>\n }<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> for \n \n $$\\{u,\\!\\hspace{0.5pt}\\hspace{0.5pt}v\\}\\in F$$<\/jats:tex-math>\n \n \n {<\/mml:mo>\n u<\/mml:mi>\n ,<\/mml:mo>\n \n \n \n v<\/mml:mi>\n }<\/mml:mo>\n \u2208<\/mml:mo>\n F<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, where set \n \n $$F\\subseteq \\left( {\\begin{array}{c}V\\\\ 2\\end{array}}\\right) $$<\/jats:tex-math>\n \n \n F<\/mml:mi>\n \u2286<\/mml:mo>\n \n \n \n \n \n V<\/mml:mi>\n <\/mml:mtd>\n <\/mml:mtr>\n \n \n \n \n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:mtd>\n <\/mml:mtr>\n <\/mml:mtable>\n <\/mml:mrow>\n <\/mml:mfenced>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is\u00a0fixed. We extend the complexity classification above by providing an explicit condition on \n \n $$(\\mu ,F)$$<\/jats:tex-math>\n \n \n (<\/mml:mo>\n \u03bc<\/mml:mi>\n ,<\/mml:mo>\n F<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> for the problem to be tractable. In order to prove the tractability part, we generalize Hirai\u2019s 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. 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