Abstract

The known linear-time kernelizations for d-Hitting Set guarantee linear worst-case running times using a quadratic-size data structure (that is not fully initialized). Getting rid of this data structure, we show that problem kernels of asymptotically optimal size O(kd) for d-Hitting Set are computable in linear time and space. Additionally, we experimentally compare the linear-time kernelizations for d-Hitting Set to each other and to a classical data reduction algorithm due to Weihe.

Highlights

  • We study data reduction algorithms for the following combinatorial optimization problem: Problem 1 (d-Hitting Set, for constant d ∈ N)

  • We describe our new contributions, which are two-fold: In Section 3, we resolve the apparent paradox that the linear worst-case running time of the known d-Hitting Set kernelizations hinges on quadratic-space data structures

  • It shows that FK and Bev work well on the very same instances, which is surprising, since they are based on different data reduction criteria and it seems that Bev should be able to apply data reduction earlier than FK

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Summary

Introduction

We study data reduction algorithms for the following combinatorial optimization problem: Problem 1 (d-Hitting Set, for constant d ∈ N). Input: A hypergraph H = (V, E) with vertex set V = {1, . The d-Hitting Set problem is an NP-complete [22] fundamental combinatorial optimization problem, arising in bioinformatics [25], medicine [24, 34], clustering [8, 21], automatic reasoning [11, 17, 31], feature selection [10, 20], radio frequency allocation [33], software engineering [29], and public transport optimization [9, 35]. Exact algorithms for NP-complete problems usually take time exponential in the input size. The main notion of data reduction with performance guarantees is kernelization [19], here stated for d-Hitting Set: Definition 2. A kernelization maps any d-Hitting Set instance (Hin, kin) to an instance (Hout, kout) in polynomial time such that

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