Linear-Time MaxCut in Multigraphs Parameterized Above the Poljak-Turzík Bound
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913410709454848 |
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| author | Lill, Jonas Petrova, Kalina Weber, Simon |
| author_facet | Lill, Jonas Petrova, Kalina Weber, Simon |
| contents | 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., $f(k)\cdot O(m+n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_01071 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Linear-Time MaxCut in Multigraphs Parameterized Above the Poljak-Turzík Bound Lill, Jonas Petrova, Kalina Weber, Simon Data Structures and Algorithms Computational Complexity Discrete Mathematics 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., $f(k)\cdot O(m+n)$. |
| title | Linear-Time MaxCut in Multigraphs Parameterized Above the Poljak-Turzík Bound |
| topic | Data Structures and Algorithms Computational Complexity Discrete Mathematics |
| url | https://arxiv.org/abs/2407.01071 |