eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-09-23
93:1
93:15
10.4230/LIPIcs.ESA.2024.93
article
Parameterized Complexity of MinCSP over the Point Algebra
Osipov, George
1
https://orcid.org/0000-0002-2884-9837
Pilipczuk, Marcin
2
https://orcid.org/0000-0001-5680-7397
Wahlström, Magnus
3
https://orcid.org/0000-0002-0933-4504
Linköping University, Sweden
University of Warsaw, Poland
Royal Holloway, University of London, UK
The input in the Minimum-Cost Constraint Satisfaction Problem (MinCSP) over the Point Algebra contains a set of variables, a collection of constraints of the form x < y, x = y, x ≤ y and x ≠ y, and a budget k. The goal is to check whether it is possible to assign rational values to the variables while breaking constraints of total cost at most k. This problem generalizes several prominent graph separation and transversal problems:
- MinCSP({<}) is equivalent to Directed Feedback Arc Set,
- MinCSP({< , ≤}) is equivalent to Directed Subset Feedback Arc Set,
- MinCSP({= ,≠}) is equivalent to Edge Multicut, and
- MinCSP({≤ ,≠}) is equivalent to Directed Symmetric Multicut. Apart from trivial cases, MinCSP({Γ}) for Γ ⊆ {< , = , ≤ ,≠} is NP-hard even to approximate within any constant factor under the Unique Games Conjecture. Hence, we study parameterized complexity of this problem under a natural parameterization by the solution cost k. We obtain a complete classification: if Γ ⊆ {< , = , ≤ ,≠} contains both ≤ and ≠, then MinCSP({Γ}) is W[1]-hard, otherwise it is fixed-parameter tractable. For the positive cases, we solve MinCSP({< , = ,≠}), generalizing the FPT results for Directed Feedback Arc Set and Edge Multicut as well as their weighted versions. Our algorithm works by reducing the problem into a Boolean MinCSP, which is in turn solved by flow augmentation. For the lower bounds, we prove that Directed Symmetric Multicut is W[1]-hard, solving an open problem.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol308-esa2024/LIPIcs.ESA.2024.93/LIPIcs.ESA.2024.93.pdf
parameterized complexity
constraint satisfaction
point algebra
multicut
feedback arc set