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Kernelization of Graph Hamiltonicity: Proper $H$-Graphs
SIAM Journal on Discrete Mathematics ( IF 0.8 ) Pub Date : 2021-04-26 , DOI: 10.1137/19m1299001
Steven Chaplick , Fedor V. Fomin , Petr A. Golovach , Dušan Knop , Peter Zeman

SIAM Journal on Discrete Mathematics, Volume 35, Issue 2, Page 840-892, January 2021.
We obtain new polynomial kernels and compression algorithms for Path Cover and Cycle Cover, the well-known generalizations of the classical Hamiltonian Path and Hamiltonian Cycle problems. Our choice of parameterization is strongly influenced by the work of Biró, Hujter, and Tuza, who in 1992 introduced $H$-graphs, intersection graphs of connected subgraphs of a subdivision of a fixed (multi-)graph $H$. In this work, we turn to proper $H$-graphs, where the containment relationship between the representations of the vertices is forbidden. As the treewidth of a graph measures how similar the graph is to a tree, the size of graph $H$ is the parameter measuring the closeness of the graph to a proper interval graph. We prove the following results. Path Cover admits a kernel of size $\mathcal{O}(\|H\|^8)$, where $\|H\|$ is the size of graph $H$. In other words, we design an algorithm that for an $n$-vertex graph $G$ and integer $k\geq 1$, in time polynomial in $n$ and $\|H\|$, outputs a graph $G'$ of size $\mathcal{O}(\|H\|^8)$ and $k'\leq |V(G')|$ such that the vertex set of $G$ is coverable by $k$ vertex-disjoint paths if and only if the vertex set of $G'$ is coverable by $k'$ vertex-disjoint paths. Hamiltonian Cycle admits a kernel of size $\mathcal{O}(\|H\|^8)$. Cycle Cover admits a polynomial kernel. We prove it by providing a compression of size $\mathcal{O}(\|H\|^{10})$ into another \sf NP-complete problem, namely, Prize Collecting Cycle Cover, that is, we design an algorithm that, in time polynomial in $n$ and $\|H\|$, outputs an equivalent instance of Prize Collecting Cycle Cover of size $\mathcal{O}(\|H\|^{10})$. In all our algorithms we assume that a proper $H$-decomposition is given as a part of the input.


中文翻译:

图哈密顿性的内核化:正确的 $H$-Graphs

SIAM 离散数学杂志,第 35 卷,第 2 期,第 840-892 页,2021 年 1 月。
我们为路径覆盖和循环覆盖获得了新的多项式内核和压缩算法,这是经典哈密顿路径和哈密顿循环问题的著名概括。我们对参数化的选择受到 Biró、Hujter 和 Tuza 工作的强烈影响,他们在 1992 年引入了 $H$-graphs,即固定(多)图 $H$ 的细分的连通子图的交集图。在这项工作中,我们转向适当的 $H$-graphs,其中顶点表示之间的包含关系是被禁止的。由于图的树宽衡量图与树的相似程度,因此图的大小 $H$ 是衡量图与适当区间图的接近程度的参数。我们证明以下结果。Path Cover 接受一个大小为 $\mathcal{O}(\|H\|^8)$ 的核,其中 $\|H\|$ 是图 $H$ 的大小。换句话说,我们设计了一个算法,对于 $n$-顶点图 $G$ 和整数 $k\geq 1$,在 $n$ 和 $\|H\|$ 中的时间多项式,输出图 $G '$ 大小为 $\mathcal{O}(\|H\|^8)$ 和 $k'\leq |V(G')|$ 使得 $G$ 的顶点集可以被 $k$ 顶点覆盖-disjoint 路径当且仅当 $G'$ 的顶点集可以被 $k'$ vertex-disjoint 路径覆盖。Hamiltonian Cycle 允许一个大小为 $\mathcal{O}(\|H\|^8)$ 的核。Cycle Cover 接受多项式核。我们通过提供一个大小为 $\mathcal{O}(\|H\|^{10})$ 的压缩到另一个 \sf NP-complete 问题,即 Prize Collecting Cycle Cover 来证明,即我们设计了一个算法在 $n$ 和 $\|H\|$ 的时间多项式中,输出大小为 $\mathcal{O}(\|H\|^{10})$ 的奖品收集循环覆盖的等效实例。
更新日期:2021-04-26
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