an:06387683
Zbl 1314.68167
van Bevern, Ren??
Towards optimal and expressive kernelization for \(d\)-hitting set
EN
Algorithmica 70, No. 1, 129-147 (2014).
00335666
2014
j
68Q25 05C65 05C85
parameterized algorithmics; linear-time data reduction; vertex cover in hypergraphs; fault diagnosis; sunflower lemma; algorithm engineering
Summary: A sunflower in a hypergraph is a set of hyperedges pairwise intersecting in exactly the same vertex set. Sunflowers are a useful tool in polynomial-time data reduction for problems formalizable as \(d\)-\textsc{Hitting Set}, the problem of covering all hyperedges (whose cardinality is bounded from above by a constant~\(d\)) of a hypergraph by at most \(k\)~vertices. Additionally, in fault diagnosis, sunflowers yield concise explanations for ``highly defective structures''.
We provide a linear-time algorithm that, by finding sunflowers, transforms an instance of \(d\)-\textsc{Hitting Set} into an equivalent instance comprising at most \(O(k^d)\)~hyperedges and vertices. In terms of parameterized complexity, we show a problem kernel with asymptotically optimal size (unless \(\operatorname {coNP}\subseteq\operatorname {NP/poly}\)) and provide experimental results that show the practical applicability of our algorithm.
Finally, we show that the number of vertices can be reduced to~\(O(k^{d-1})\) with additional processing in \(O(k^{1.5d})\)~time -- nontrivially combining the sunflower technique with problem kernels due to Abu-Khzam and Moser.