Parameterized Complexity of MinCSP over the Point Algebra
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.
parameterized complexity
constraint satisfaction
point algebra
multicut
feedback arc set
Theory of computation~Fixed parameter tractability
93:1-93:15
Regular Paper
This work was carried out during the Copenhagen Summer of Counting & Algebraic Complexity, funded by research grants from VILLUM FONDEN (Young Investigator Grant 53093) and the European Union (ERC, CountHom, 101077083). Views and opinions expressed are those of the authors only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.
We thank the anonymous reviewers whose detailed feedback significantly improved presentation of the paper.
George
Osipov
George Osipov
Linköping University, Sweden
https://orcid.org/0000-0002-2884-9837
George was supported by the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation.
Marcin
Pilipczuk
Marcin Pilipczuk
University of Warsaw, Poland
https://orcid.org/0000-0001-5680-7397
During this research Marcin was part of BARC, supported by the VILLUM Foundation grant 16582.
Magnus
Wahlström
Magnus Wahlström
Royal Holloway, University of London, UK
https://orcid.org/0000-0002-0933-4504
10.4230/LIPIcs.ESA.2024.93
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George Osipov, Marcin Pilipczuk, and Magnus Wahlström
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