394 research outputs found
Concentration inequalities for the number of real zeros of Kac polynomials
We study concentration inequalities for the number of real roots of the
classical Kac polynomials where
are independent random variables with mean 0, variance 1, and uniformly bounded
(2+\ep_0)-moments. We establish polynomial tail bounds, which are optimal,
for the bulk of roots. For the whole real line, we establish sub-optimal tail
bounds.Comment: more references adde
Complexity of the (Connected) Cluster Vertex Deletion problem on -free graphs
The well-known Cluster Vertex Deletion problem (CVD) asks for a given graph
and an integer whether it is possible to delete a set of at most
vertices of such that the resulting graph is a cluster graph (a
disjoint union of cliques). We give a complete characterization of graphs
for which CVD on -free graphs is polynomially solvable and for which it is
NP-complete. Moreover, in the NP-completeness cases, CVD cannot be solved in
sub-exponential time in the vertex number of the -free input graphs unless
the Exponential-Time Hypothesis fails. We also consider the connected variant
of CVD, the Connected Cluster Vertex Deletion problem (CCVD), in which the set
has to induce a connected subgraph of . It turns out that CCVD admits
the same complexity dichotomy for -free graphs. Our results enlarge a list
of rare dichotomy theorems for well-studied problems on -free graphs.Comment: Extended version of a MFCS 2022 paper. To appear in Theory of
Computing System
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