Linear-Time MaxCut in Multigraphs Parameterized Above the Poljak-Turzík Bound
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Date
2024
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Conference Paper
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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., $f(k)\cdot O(m)$. 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., $f(k)\cdot O(m+n)$.
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Book title
19th International Symposium on Parameterized and Exact Computation (IPEC 2024)
Volume
321
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Publisher
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Event
19th International Symposium on Parameterized and Exact Computation (IPEC 2024)
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Subject
Fixed-parameter tractability; maximum cut; Edwards-Erdős bound; Poljak-Turzík bound; multigraphs; integer-weighted graphs
Organisational unit
03672 - Steger, Angelika / Steger, Angelika
08817 - Gärtner, Bernd (Tit.-Prof.) / Gärtner, Bernd (Tit.-Prof.)
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Funding
173721 - Temporal Information Integration in Neural Networks (SNF)
204320 - Unique Sink Orientations (SNF)
204320 - Unique Sink Orientations (SNF)