Budgeted Dominating Sets in Uncertain Graphs
Keerti Choudhary, Avi Cohen, N. S. Narayanaswamy, David Peleg, R., Vijayaragunathan

TL;DR
This paper investigates the computational complexity and approximation algorithms for the Probabilistic Budgeted Dominating Set problem in uncertain graphs, revealing NP-hardness, parameterized hardness, and efficient solutions in special cases.
Contribution
It establishes the NP-completeness of PBDS on trees, explores its parameterized complexity, and provides approximation schemes and exact algorithms for specific graph classes.
Findings
PBDS is NP-complete on trees of diameter at most four.
PBDS is W[1]-hard for the budget parameter k.
A PTAS exists for PBDS on trees with (1-ε) approximation.
Abstract
We study the {\em Budgeted Dominating Set} (BDS) problem on uncertain graphs, namely, graphs with a probability distribution associated with the edges, such that an edge exists in the graph with probability . The input to the problem consists of a vertex-weighted uncertain graph and an integer {\em budget} (or {\em solution size}) , and the objective is to compute a vertex set of size that maximizes the expected total domination (or total weight) of vertices in the closed neighborhood of . 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…
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