2 research outputs found
Streaming Kernelization
Kernelization is a formalization of preprocessing for combinatorially hard
problems. We modify the standard definition for kernelization, which allows any
polynomial-time algorithm for the preprocessing, by requiring instead that the
preprocessing runs in a streaming setting and uses
bits of memory on instances . We obtain
several results in this new setting, depending on the number of passes over the
input that such a streaming kernelization is allowed to make. Edge Dominating
Set turns out as an interesting example because it has no single-pass
kernelization but two passes over the input suffice to match the bounds of the
best standard kernelization
Towards Optimal and Expressive Kernelization for d-Hitting Set
d-Hitting Set is the NP-hard problem of selecting at most k vertices of a
hypergraph so that each hyperedge, all of which have cardinality at most d,
contains at least one selected vertex. The applications of d-Hitting Set are,
for example, fault diagnosis, automatic program verification, and the
noise-minimizing assignment of frequencies to radio transmitters.
We show a linear-time algorithm that transforms an instance of d-Hitting Set
into an equivalent instance comprising at most O(k^d) hyperedges and vertices.
In terms of parameterized complexity, this is a problem kernel. Our
kernelization algorithm is based on speeding up the well-known approach of
finding and shrinking sunflowers in hypergraphs, which yields problem kernels
with structural properties that we condense into the concept of expressive
kernelization.
We conduct experiments to show that our kernelization algorithm can kernelize
instances with more than 10^7 hyperedges in less than five minutes.
Finally, we show that the number of vertices in the problem kernel can be
further reduced to O(k^{d-1}) with additional O(k^{1.5 d}) processing time by
nontrivially combining the sunflower technique with d-Hitting Set problem
kernels due to Abu-Khzam and Moser.Comment: This version gives corrected experimental results, adds additional
figures, and more formally defines "expressive kernelization