Minimum 0-extension problems on directed metrics

For a metric μ on finite set T, the minimum 0-extension problem 0-Ext[μ] is defined as follows: Given V⊇T and c:V2→Q+, minimize ∑c(xy)μ(γ(x),γ(y)) subject to γ:V→T,γ(t)=t(∀t∈T), where sum taken over all unordered pairs in V. This generalizes several classical combinatorial optimization problems such cut or multiterminal problem. Karzanov Hirai established complete classification of metrics for which polynomial time solvable NP-hard. result can also be viewed sharpening general dichotomy theorem finite-valued CSPs (Thapper Živný 2016) specialized 0-Ext[μ]. In this paper, we consider directed version 0⃗-Ext[μ] problem, c are not assumed symmetric. We extend NP-hardness condition 0⃗-Ext[μ]: If cannot represented shortest path an orientable modular graph with orbit-invariant “directed” edge-length, then show partial converse: lattice tractable. further provide new characteristic 0⃗-Ext[μ], establish case star.