关于紧下界上的最大二等分的注记
Note on Maximal Bisection above Tight Lower Bound
摘要：
在图$G=(V,E)$中，二分$(X,Y)$是将$V$划分为集合$X$和$Y$，使得$|X|\le |Y|\le |X|+1$。$(X,Y)$的大小是$X$和$Y$之间的边数。在最大二分法问题中，我们给出一个图$G=(V,E)$，并要求找到一个最大尺寸的二分法。不难看出，$\lceil |E|/2 \rceil$是$G$二等分的最大尺寸的紧下界。本文研究了以下参数化问题的参数化复杂性，称为紧下界上的最大二分法（Max-Bisec-ATLB）：判定图$G=(V,E)$是否有大小至少为$\lceil |E|/2 \rceil+k,$的二分法，其中$k$是参数.我们表明，这个参数化的问题有一个核心与$O(k^2)$顶点和$O(k^3)$边，即，Max-Bisec-ATLB的每个实例等价于具有至多$O(k^2)$顶点和$O(k^3)$边的图上的Max-Bisec-ATLB的实例。
英文摘要：
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.
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