Given a positive integer $d$, the $d$-CUT problem is to decide if an undirected graph $G=(V,E)$ has a non trivial bipartition $(A,B)$ of $V$ such that every vertex in $A$ (resp. $B$) has at most $d$ neighbors in $B$ (resp. $A$). When $d=1$, this is the MATCHING CUT problem. Gomes and Sau, in IPEC 2019, gave the first fixed parameter tractable algorithm for $d$-CUT, when parameterized by maximum number of the crossing edges in the cut (i.e. the size of edge cut). However, their paper doesn't provide an explicit bound on the running time, as it indirectly relies on a MSOL formulation and Courcelle's Theorem. Motivated by this, we design and present an FPT algorithm for the MATCHING CUT (and more generally for $d$-CUT) for general graphs with running time $2^{O(k\log k)}n^{O(1)}$ where $k$ is the maximum size of the edge cut. This is the first FPT algorithm for the MATCHING CUT (and $d$-CUT) with an explicit dependence on this parameter. We also observe a lower bound of $2^{\Omega(k)}n^{O(1)}$ with same parameter for MATCHING CUT assuming ETH.
翻译:鉴于美元整数正整数,美元-CUT问题在于决定一个非方向图$G=(V,E)美元是否具有非微不足道的双向双偏差(A,B)美元(美元)美元(美元),这样,美元(Rest. $B$)的每个顶点最多以美元为美元左右。当美元=1美元时,这是Matching CUT问题。在IPEC 2019 中,Gomes和Sae(Gomes和Saau)首次给出了美元-CUT的固定参数可拉动算法,当用削减中最大跨越边缘数(A,B)美元(美元)的比值(A,B)美元,这样他们的文件并没有在运行时提供明确的约束,因为它间接依赖MSOL的配方和Courcelle的理论。受此驱动,我们设计并提出了Matching CUT(G) (更一般为美元-C-CUT) 通用算法,用于运行总图的时间为2°(k) 美元/克_Q) 基) 的比值最大直值(MAUTUT) 的比值(C) 的比值(C) 的比值最大。