MaxCut is a classical NP-complete problem and a crucial building block in many combinatorial algorithms. The famous Edwards-Erd\H{o}s 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-Erd\H{o}s 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-Erd\H{o}s bound, we use the difference to the Poljak-Turz\'ik bound. The Poljak-Turz\'ik 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-Turz\'ik bound can be larger and thus yield a smaller parameter k. Our algorithm also runs in parameterized linear time, i.e., $f(k)\cdot O(m+n)$.

翻译：最大割（MaxCut）是一个经典的NP完全问题，也是许多组合算法中的关键构建模块。著名的Edwards-Erdős界指出：任何具有n个顶点和m条边的连通图都包含一个大小至少为$m/2 + (n-1)/4$的割。Crowston、Jones和Mnich [Algorithmica, 2015]证明了在简单连通图上的最大割问题存在一个FPT算法，其中参数k是期望割大小c与Edwards-Erdős界给出的下界之间的差值。随后，Etscheid和Mnich [Algorithmica, 2017]将该算法改进至参数化线性时间运行，即$f(k)\cdot O(m)$。我们从两个方面改进了这一结果：首先，我们将算法扩展至适用于多重图（或具有正整数权重的图）。其次，我们改变了参数设置；不再使用与Edwards-Erdős界的差值，而是采用与Poljak-Turzík界的差值作为参数。Poljak-Turzík界指出：任何加权图G都存在一个大小至少为$w(G)/2 + w_{MSF}(G)/4$的割，其中w(G)表示G的总权重，$w_{MSF}(G)$表示其最小生成森林的权重。在连通简单图中，这两个界是等价的，但对于多重图，Poljak-Turzík界可能更大，从而产生更小的参数k。我们的算法同样在参数化线性时间内运行，即$f(k)\cdot O(m+n)$。