Nearly Time-Optimal Kernelization Algorithms for the Line-Cover Problem with Big Data

Abstract

Based on well-known complexity theory conjectures, any polynomial-time kernelization algorithm for the NP-hard Line-Cover problem produces a kernel of size \(\Omega (k^2)\), where k is the size of the sought line cover. Motivated by the current research in massive data processing, we study the existence of kernelization algorithms with limited space and time complexity for Line-Cover. We prove that every kernelization algorithm for Line-Cover takes time \(\Omega (n \log k + k^2 \log k)\), and present a randomized kernelization algorithm for Line-Cover that produces a kernel of size bounded by \(k^2\), and runs in time \({\mathcal {O}}(n \log k + k^2 (\log k \log \log k)^2)\) and space \({\mathcal {O}}(k^2\log ^{2} k)\). Our techniques are also useful for developing deterministic kernelization algorithms for Line-Cover with limited space and improved running time, and for developing streaming kernelization algorithms for Line-Cover with near-optimal update-time.