Tight Running Time Lower Bounds for Vertex Deletion Problems
release_zaembf6esvawvdb55ofqak65uu
by
Christian Komusiewicz
2015
Abstract
For a graph class Π, the Π-Vertex Deletion problem has as input an
undirected graph G=(V,E) and an integer k and asks whether there is a set
of at most k vertices that can be deleted from G such that the resulting
graph is a member of Π. By a classic result of Lewis and Yannakakis [J.
Comput. Syst. Sci. '80], Π-Vertex Deletion is NP-hard for all hereditary
properties Π. We adapt the original NP-hardness construction to show that
under the Exponential Time Hypothesis (ETH) tight complexity results can be
obtained. We show that Π-Vertex Deletion does not admit a 2^o(n)-time
algorithm where n is the number of vertices in G. We also obtain a
dichotomy for running time bounds that include the number m of edges in the
input graph: On the one hand, if Π contains all independent sets, then
there is no 2^o(n+m)-time algorithm for Π-Vertex Deletion. On the other
hand, if there is a fixed independent set that is not contained in Π and
containment in Π can determined in 2^O(n) time or 2^o(m) time, then
Π-Vertex Deletion can be solved in 2^O(√(m))+O(n) or
2^o(m)+O(n) time, respectively. We also consider restrictions on the
domain of the input graph G. For example, we obtain that Π-Vertex
Deletion cannot be solved in 2^o(√(n)) time if G is planar and Π
is hereditary and contains and excludes infinitely many planar graphs. Finally,
we provide similar results for the problem variant where the deleted vertex set
has to induce a connected graph.
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