eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-18
46:1
46:13
10.4230/LIPIcs.MFCS.2020.46
article
Minimum 0-Extension Problems on Directed Metrics
Hirai, Hiroshi
1
https://orcid.org/0000-0002-4784-5110
Mizutani, Ryuhei
1
https://orcid.org/0000-0003-2944-9066
Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo, Japan
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:(V 2) → 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. The complexity dichotomy of 0-Ext[μ] was established by Karzanov and Hirai, which is viewed as a manifestation of the dichotomy theorem for finite-valued CSPs due to Thapper and Živný.
In this paper, we consider a directed version 0→-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 0→-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 0→-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 0→-Ext[μ] is tractable. We further provide a new NP-hardness condition characteristic of 0→-Ext[μ], and establish a dichotomy for the case where μ is a directed metric of a star.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol170-mfcs2020/LIPIcs.MFCS.2020.46/LIPIcs.MFCS.2020.46.pdf
Minimum 0-extension problems
Directed metrics
Valued constraint satisfaction problems
Computational complexity