Note on Maximal Bisection above Tight Lower Bound

In a graph $G=(V,E)$, a bisection $(X,Y)$ is a partition of $V$ into sets $X$ and $Y$ such that $|X|\le |Y|\le |X|+1$. The size of $(X,Y)$ is the number of edges between $X$ and $Y$. In the Max Bisection problem we are given a graph $G=(V,E)$ and are required to find a bisection of maximum size. It is not hard to see that $\lceil |E|/2 \rceil$ is a tight lower bound on the maximum size of a bisection of $G$. We study parameterized complexity of the following parameterized problem called Max Bisection above Tight Lower Bound (Max-Bisec-ATLB): decide whether a graph $G=(V,E)$ has a bisection of size at least $\lceil |E|/2 \rceil+k,$ where $k$ is the parameter. We show that this parameterized problem has a kernel with $O(k^2)$ vertices and $O(k^3)$ edges, i.e., every instance of Max-Bisec-ATLB is equivalent to an instance of Max-Bisec-ATLB on a graph with at most $O(k^2)$ vertices and $O(k^3)$ edges.

💡 Research Summary

The paper investigates the Max‑Bisection problem from the perspective of parameterization above a tight lower bound. In a graph G = (V,E), a bisection (X,Y) partitions the vertex set into two parts whose sizes differ by at most one, and its size is the number of edges crossing between X and Y. It is easy to see that any bisection obtained by placing each vertex independently at random yields an expected crossing size of |E|/2, which implies that ⌈|E|/2⌉ is a universal lower bound on the optimum. Moreover, this bound is tight: there exist graphs for which the optimum equals the bound.

The authors define the parameterized decision problem Max‑Bisec‑ATLB (Max‑Bisection Above Tight Lower Bound): given a graph G and an integer k, decide whether G admits a bisection of size at least ⌈|E|/2⌉ + k, where k is the parameter. The goal is to understand the fixed‑parameter tractability of this problem with respect to k.

The main technical contribution is a kernelization algorithm that reduces any instance (G,k) to an equivalent instance (G′,k) whose size is bounded by a polynomial in k. Two reduction rules are introduced:

1. High‑Degree Vertex Rule – If a vertex v has degree at least 2k + 1, then v can be safely removed. All edges incident to v are deleted, and for each pair of neighbors u,w of v a new edge (u,w) with appropriate weight is added. The intuition is that any bisection can gain at most k extra crossing edges from v, so removing v does not affect the existence of a solution above the bound. Repeated application guarantees that every remaining vertex has degree at most 2k.

2. Edge‑Count Rule – After the first rule, the graph contains at most O(k²) vertices (since each vertex contributes at most 2k edges). Consequently the total number of edges is bounded by O(k³). If the edge count exceeds this bound, the instance is immediately a “yes” instance because the surplus edges guarantee a crossing of at least ⌈|E|/2⌉ + k.

Both rules can be applied in polynomial time (linear in the size of the input). After exhaustive reduction, the resulting kernel has at most O(k²) vertices and O(k³) edges. This establishes that Max‑Bisec‑ATLB is fixed‑parameter tractable and admits a polynomial kernel of the stated size.

The paper situates this result within the broader literature on parameterization above or below guaranteed values, a line of research that includes Max‑Cut above the Edwards–Erdős bound and other graph partition problems. By focusing on the natural lower bound ⌈|E|/2⌉, the authors provide a clean and conceptually simple parameterization that yields a relatively small kernel compared with many earlier works.

In the discussion, the authors point out several avenues for future research. One direction is to improve the kernel size, possibly reducing the vertex bound from O(k²) to O(k) or achieving a linear‑edge kernel. Another is to extend the “above tight lower bound” framework to related problems such as Max‑Cut, Minimum Bisection, or multipartite partition problems. Finally, they suggest investigating whether the kernelization techniques can be combined with exact exponential‑time algorithms to obtain improved running times for moderate values of k.

Overall, the paper contributes a clear kernelization result for a natural parameterization of Max‑Bisection, demonstrating that the problem is fixed‑parameter tractable with respect to the excess over the trivial lower bound and providing a concrete polynomial‑size kernel that may serve as a building block for further algorithmic developments.