06.09.2022
Target Set Selection Parameterized by Vertex Cover and More
Erschienen in: Theory of Computing Systems | Ausgabe 5/2022
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Abstract
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It was shown by Nichterlein et al. (Soc. Netw. Anal. Min. 3(2), 233–256 10) that it is possible to compute an optimal-sized target set in \(\boldsymbol {O}(\boldsymbol {2}^{(\boldsymbol {2}^{t}+\boldsymbol {1})\boldsymbol {t}}\boldsymbol {\cdot m})\) time, where t denotes the cardinality of a minimum degree-0 modulator of G. We improve this result by designing an algorithm running in time \(\boldsymbol {2}^{\boldsymbol {O}(\boldsymbol {t}\log \boldsymbol {t})}\boldsymbol {n}\).
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We design a \(\boldsymbol {2}^{\boldsymbol {2}^{\boldsymbol {O}(\boldsymbol {t})}}\boldsymbol {n}^{\boldsymbol {O}(\boldsymbol {1})}\) time algorithm to compute an optimal target set for G, where t is the size of a minimum degree-1 modulator of G.
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We show that for a graph on n vertices of treewidth s, the TSS problem cannot be solved in \(\boldsymbol {f}(\boldsymbol {s})\boldsymbol {n}^{\boldsymbol {o}(\boldsymbol {\frac {s}{\log s}})}\) time unless the Exponential Time Hypothesis fails. This is an improvement over the previously known lower bound of \(\boldsymbol {f}(\boldsymbol {s})\boldsymbol {n}^{\boldsymbol {o}(\sqrt {\boldsymbol {s}})}\) due to Ben-Zwi et al. (Discret. Optim. 8(1), 87–96 16). In fact, we prove that same lower bound holds when parameterized by tree-depth or star-deletion number.