Subexponential fixed-parameter algorithms for partial vector domination

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• 作者：Toshimasa Ishii^a ; ^{ishii@econ.hokudai.ac.jp" class="auth_mail" title="E-mail the corresponding author} ; Hirotaka Ono^b ; ^{hirotaka@econ.kyushu-u.ac.jp" class="auth_mail" title="E-mail the corresponding author} ; Yushi Uno^c ; ^{uno@mi.s.osakafu-u.ac.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：(Total) vector dominating set ; Partial dominating set ; Fixed-parameter tractability ; Branchwidth ; Apex-minor-free graphs
• 刊名：Discrete Optimization
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：22
• 期：part_PA
• 页码：111-121
• 全文大小：652 K

文摘

Given a graph G=(V,E)$G=(V,E)$ of order n$n$ and an n$n$-dimensional non-negative vector $d=(d\left(1\right),d\left(2\right),\dots ,d\left(n\right))$, called demand vector, the vector domination (resp., total vector domination) is the problem of finding a minimum S⊆V$S\subseteq V$ such that every vertex v$v$ in V∖S$V\setminus S$ (resp., in ac7dac9c2e48cc9f1808ebf36009" title="Click to view the MathML source">V$V$) has at least d(v)$d\left(v\right)$ neighbors in S$S$. The (total) vector domination is a generalization of many dominating set type problems, e.g., the (total) dominating set problem, the (total) k$k$-dominating set problem (this k$k$ is different from the solution size), and so on, and subexponential fixed-parameter algorithms with respect to solution size for apex-minor-free graphs (so for planar graphs) are known. In this paper, we consider maximization versions of the problems; that is, for a given integer k$k$, the goal is to find an S⊆V$S\subseteq V$ with size k$k$ that maximizes the total sum of satisfied demands. For these problems, we design subexponential fixed-parameter algorithms with respect to k$k$ for apex-minor-free graphs.