Budgeted Dominating Sets in Uncertain Graphs

We study the 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 budget (or 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 Probabilistic Budgeted Dominating Set (PBDS) problem. In this article, we present the following results on the complexity of the PBDS problem. 1. We show that the PBDS problem is NP-complete even when restricted to uncertain 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 W[1]-hard for the budget parameter k, and under the Exponential time hypothesis it cannot be solved in no(k) time. 2. We show that if one is willing to settle for (1 ) 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. 3. We consider the parameterized complexity of the PBDS problem, and show that Uni-PBDS (where all edge probabilities are identical) is W[1]-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. 4. Finally, we extend some of our parameterized results to planar and apex-minor-free graphs. Our first hardness proof (Thm. 1) makes use of the new problem of k-Subset ΣΠ Maximization (k-SPM), which we believe is of independent interest. We prove its NP-hardness by a reduction from the well-known k-SUM problem, presenting a close relationship between the two problems.