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 $\G=(V, E, p, \omega)$ 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 $(1-\epsilon)$ 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}
翻译:我们在不确定的图表中研究 { 预算的 Dominate Set} (BDS) (BDS) 问题, 即 以概率分配 $p$ 的图形, 与边缘相关, 从而在图形中存在 美元, 概率 $ p(e) 美元 。 给问题的投入包括一个顶点加权的不确定图形$\ G=( V, E, p,, \ omga) 和一个整数 {emplio } (或 平流大小}) $k$, 目标是计算一个以美元为单位的顶点设置 $ $ (S) 美元, 使关闭的周围的 PSDR 的预期总体支配( 或总重量) 。 我们的 PBDSDS 问题, 也存在所有问题, 而当我们限制直径直的 美元 美元 美元 。 这和 直径的直径直径比 直的 RDFDS 的直径比 是一个明的直径比 。