Linear-Time MaxCut in Multigraphs Parameterized Above the Poljak-Turzík Bound

MaxCut is a classical \(\textsf{NP}\)-complete problem and a crucial building block in many combinatorial algorithms. The famous Edwards-Erdös bound states that any connected graph on n vertices with m edges contains a cut of size at least \(\frac{m}{2}+\frac{n-1}{4}\). 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-Erdös 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-Erdös bound, we use the difference to the Poljak-Turzík bound. The Poljak-Turzík bound states that any weighted graph G has a cut of weight at least \(\frac{w(G)}{2}+\frac{w_{MSF}(G)}{4}\), 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., \(f(k)\cdot O(m+n)\).