2 research outputs found
A Generalization of Nemhauser and Trotter's Local Optimization Theorem
The Nemhauser-Trotter local optimization theorem applies to the NP-hard
Vertex Cover problem and has applications in approximation as well as
parameterized algorithmics. We present a framework that generalizes Nemhauser
and Trotter's result to vertex deletion and graph packing problems, introducing
novel algorithmic strategies based on purely combinatorial arguments (not
referring to linear programming as the Nemhauser-Trotter result originally
did). We exhibit our framework using a generalization of Vertex Cover, called
Bounded- Degree Deletion, that has promise to become an important tool in the
analysis of gene and other biological networks. For some fixed d \geq 0,
Bounded-Degree Deletion asks to delete as few vertices as possible from a graph
in order to transform it into a graph with maximum vertex degree at most d.
Vertex Cover is the special case of d = 0. Our generalization of the
Nemhauser-Trotter theorem implies that Bounded-Degree Deletion has a problem
kernel with a linear number of vertices for every constant d. We also outline
an application of our extremal combinatorial approach to the problem of packing
stars with a bounded number of leaves. Finally, charting the border between
(parameterized) tractability and intractability for Bounded-Degree Deletion, we
provide a W[2]-hardness result for Bounded-Degree Deletion in case of unbounded
d-values