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Perfectly Matched Sets in Graphs: Hardness, Kernelization Lower Bound, and FPT and Exact Algorithms
arXiv - CS - Discrete Mathematics Pub Date : 2021-07-19 , DOI: arxiv-2107.08584
N. R. Aravind, Roopam Saxena

In an undirected graph $G=(V,E)$, we say $(A,B)$ is a pair of perfectly matched sets if $A$ and $B$ are disjoint subsets of $V$ and every vertex in $A$ (resp. $B$) has exactly one neighbor in $B$ (resp. $A$). The size of a pair perfectly matched sets $(A,B)$ is $|A|=|B|$. The PERFECTLY MATCHED SETS problem is to decide whether a given graph $G$ has a pair of perfectly matched sets of size $k$. We show that PMS is $W[1]$-hard when parameterized by solution size $k$ even when restricted to split graphs and bipartite graphs. We give FPT algorithms with respect to the parameters distance to cluster, distance to co-cluster and treewidth. We also provide an exact exponential algorithm running in time $O^*(1.964^n)$.

中文翻译:

图中的完美匹配集:硬度、核化下界、FPT 和精确算法

在无向图 $G=(V,E)$ 中,如果 $A$ 和 $B$ 是 $V$ 的不相交子集并且 $(A,B)$ 是一对完全匹配的集合,并且 $ A$ (resp. $B$) 在 $B$ (resp. $A$) 中正好有一个邻居。一对完美匹配集 $(A,B)$ 的大小是 $|A|=|B|$。完美匹配集问题是确定给定的图 $G$ 是否有一对大小为 $k$ 的完美匹配集。我们表明,当由解决方案大小 $k$ 参数化时,PMS 是 $W[1]$-hard,即使仅限于分割图和二分图。我们给出了关于聚类距离、共同聚类距离和树宽参数的 FPT 算法。我们还提供了一个在时间 $O^*(1.964^n)$ 中运行的精确指数算法。
更新日期:2021-07-20
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