不确定图的扩张控制集
Budgeted Dominating Sets in Uncertain Graphs
摘要：
本文研究了不确定图上的{ \em支配集}（BDS）问题，即边的概率分布为$p$的图，使得边$e$以概率$p(e)$存在于图中.问题的输入由顶点加权不确定图$\G=(V, E, p, ω)$和整数{ \em预算}（或{ \em解大小}）$k$组成，目标是计算大小为$k$的顶点集$S$，使$S$的闭邻域中顶点的期望总支配（或总权重）最大化.我们把这个问题称为{ \em概率支配支配集}~（PBDS）问题，并给出了以下结果. \begin{enumerate} \dnsitem We show that the PBDS problem is NP-complete even when restricted to uncertain {\em trees} of diameter at most four. This is in sharp contrast with the well-known fact that the BDS problem is solvable in polynomial time in trees. We further show that PBDS is \wone-hard for the budget parameter $k$, and under the {\em Exponential time hypothesis} it cannot be solved in $n^{o(k)}$ time. \item We show that if one is willing to settle for $(1-ε)$ approximation, then there exists a PTAS for PBDS on trees. Moreover, for the scenario of uniform edge-probabilities, the problem can be solved optimally in polynomial time. \item We consider the parameterized complexity of the PBDS problem, and show that Uni-PBDS (where all edge probabilities are identical) is \wone-hard for the parameter pathwidth. On the other hand, we show that it is FPT in the combined parameters of the budget $k$ and the treewidth. \item Finally, we extend some of our parameterized results to planar and apex-minor-free graphs. \end{enumerate}
英文摘要：
We study the {\em Budgeted Dominating Set} (BDS) problem on uncertain graphs, namely, graphs with a probability distribution $p$ associated with the edges, such that an edge $e$ exists in the graph with probability $p(e)$. The input to the problem consists of a vertex-weighted uncertain graph $\G=(V, E, p, ω)$ and an integer {\em budget} (or {\em solution size}) $k$, and the objective is to compute a vertex set $S$ of size $k$ that maximizes the expected total domination (or total weight) of vertices in the closed neighborhood of $S$. We refer to the problem as the {\em Probabilistic Budgeted Dominating Set}~(PBDS) problem and present the following results. \begin{enumerate} \dnsitem We show that the PBDS problem is NP-complete even when restricted to uncertain {\em trees} of diameter at most four. This is in sharp contrast with the well-known fact that the BDS problem is solvable in polynomial time in trees. We further show that PBDS is \wone-hard for the budget parameter $k$, and under the {\em Exponential time hypothesis} it cannot be solved in $n^{o(k)}$ time. \item We show that if one is willing to settle for $(1-ε)$ approximation, then there exists a PTAS for PBDS on trees. Moreover, for the scenario of uniform edge-probabilities, the problem can be solved optimally in polynomial time. \item We consider the parameterized complexity of the PBDS problem, and show that Uni-PBDS (where all edge probabilities are identical) is \wone-hard for the parameter pathwidth. On the other hand, we show that it is FPT in the combined parameters of the budget $k$ and the treewidth. \item Finally, we extend some of our parameterized results to planar and apex-minor-free graphs. \end{enumerate}
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