Towards Optimal and Expressive Kernelization for d-Hitting Set

TL;DR

This paper introduces a fast, expressive kernelization algorithm for d-Hitting Set that significantly reduces problem size, enabling efficient processing of large instances with practical applications.

Contribution

It presents a novel linear-time kernelization method based on sunflower techniques, achieving smaller kernels and improved efficiency for d-Hitting Set.

Findings

Kernelization reduces instances to O(k^d) hyperedges and vertices.

Algorithm can handle over 10^7 hyperedges in under five minutes.

Further reduction to O(k^{d-1}) vertices is possible with additional processing.

Abstract

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.