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Linear-Time MaxCut in Multigraphs Parameterized Above the Poljak-Turzík Bound
Authors
Kalina Petrova 
,
Simon Weber
-
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19th International Symposium on Parameterized and Exact Computation (IPEC 2024)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: International Symposium on Parameterized and Exact Computation (IPEC) - License:
Creative Commons Attribution 4.0 International license
- Publication Date: 2024-12-05
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Jonas Lill, Kalina Petrova, and Simon Weber. Linear-Time MaxCut in Multigraphs Parameterized Above the Poljak-Turzík Bound. In 19th International Symposium on Parameterized and Exact Computation (IPEC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 321, pp. 2:1-2:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.IPEC.2024.2
Abstract
MaxCut is a classical NP-complete problem and a crucial building block in many combinatorial algorithms. The famous Edwards-Erdős bound states that any connected graph on n vertices with m edges contains a cut of size at least m/2+(n-1)/4. Crowston, Jones and Mnich [Algorithmica, 2015] showed that the MaxCut problem on simple connected graphs admits an FPT algorithm, where the parameter k is the difference between the desired cut size c and the lower bound given by the Edwards-Erdős bound. This was later improved by Etscheid and Mnich [Algorithmica, 2017] to run in parameterized linear time, i.e., f(k)⋅ O(m). We improve upon this result in two ways: Firstly, we extend the algorithm to work also for multigraphs (alternatively, graphs with positive integer weights). Secondly, we change the parameter; instead of the difference to the Edwards-Erdős bound, we use the difference to the Poljak-Turzík bound. The Poljak-Turzík bound states that any weighted graph G has a cut of size at least (w(G))/2+(w_MSF(G))/4, where w(G) denotes the total weight of G, and w_MSF(G) denotes the weight of its minimum spanning forest. In connected simple graphs the two bounds are equivalent, but for multigraphs the Poljak-Turzík bound can be larger and thus yield a smaller parameter k. Our algorithm also runs in parameterized linear time, i.e., f(k)⋅ O(m+n).
Subject Classification
ACM Subject Classification
- Mathematics of computing → Graph algorithms
- Mathematics of computing → Combinatorial optimization
- Theory of computation → Fixed parameter tractability
Keywords
- Fixed-parameter tractability
- maximum cut
- Edwards-Erdős bound
- Poljak-Turzík bound
- multigraphs
- integer-weighted graphs
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References
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