Minimum 0-extension problems on directed metrics

Abstract

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