Streaming Kernelization

Abstract

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 O(poly(k) log |x|) bits of memory on instances (x, k). 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.⋆ Supported by the Emmy Noether-program of the DFG, KR 4286/1. 1 Unless NP ⊆ coNP/poly and the polynomial hierarchy collapses.

Kernelization via Sampling with Applications to Finding Matchings and Related Problems in Dynamic Graph Streams

“…A simple observation shows that any lower bound for parameterized streaming kernels also transfers for the parameterized streaming algorithms. Thus the results of Fafiane and Kratsch [19] also give lower bounds for the parameterized streaming algorithms for these problems. However, our lower bounds have the following advantage over the results of [19]:…”

“…Fafianie and Kratsch [19] gave lower bounds for several parameterized problems. In particular, they showed that: • Any t-pass kernel for CLUSTER EDITING(k) and MINIMUM FILL-IN(k) requires Ω(n/t) space.…”

“…The concept of parameterized stream algorithms was explored by Chitnis et al [11] and Fafianie and Kratsch [19]. Their work investigated a natural connection between data streams and parameterized complexity.…”

Dynamic Graph Stream Algorithms in o(n) Space

“…To the best of our knowledge, very few results are known in this direction. Chitnis et al [CCHM15] and Fafianie and Kratsch [FK14] introduced parameterized graph stream algorithms which may only use o(n) space with some promise of the size of the solution. This parameterized setting has been further investigated in [CCE + 16].…”

A linear-time kernelization for the Rooted k-Leaf Outbranching Problem

“…Another measure of the performance of a kernelization is the number of passes over the input the kernelization makes [11].…”