Linear-Time MaxCut in Multigraphs Parameterized Above the Poljak-Turz\'ik Bound

MaxCut is a classical NP-complete problem and a crucial building block in many combinatorial algorithms. The famous Edwards-Erdos 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-Erdos 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-Erdos 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₌ₒ₅ (G) /4, where w (G) denotes the total weight of G, and w₌ₒ₅ (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) O (m+n).