Linear-Time MaxCut in Multigraphs Parameterized Above the Poljak-Turzík Bound
Jonas Lill, Kalina Petrova, Simon Weber
TL;DR
This work extends the fixed-parameter tractability of MaxCut above guaranteed lower bounds from simple graphs to multigraphs and positive-weighted graphs by leveraging the Poljak-Turzík bound $\mu(G) \ge \frac{w(G)}{2}+\frac{w_{MSF}(G)}{4}$. It presents a parameterized-linear-time algorithm for deciding MaxCut above this bound and shows how to compute such a cut efficiently, including a parameterized quadratic-time variant and a $2^{O(k)}$-factorized algorithm with time $O(|E||V|)$. The technique relies on a refined set of reduction rules that reduce the instance to a small separator $S$ with $|S|=O(k)$, after which the remaining graph $G-S$ is a uniform-clique-forest and partitions of $S$ are exhaustively explored, using a vertex-weighted MaxCut reduction on $G-S$. By extending the bound from Edwards-Erdős to Poljak-Turzík and broadening the graph class, the paper achieves faster exact algorithms for a broader family of MaxCut instances, with clear implications for exact optimization on weighted and multigraph inputs.
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., $f(k)\cdot O(m+n)$.