Computing Kernels in Parallel: Lower and Upper Bounds

TL;DR

This paper investigates the parallel computability of problem kernels, revealing complex trade-offs between kernel size and circuit depth, with some kernels efficiently computable in parallel and others inherently sequential.

Contribution

It establishes upper and lower bounds on the parallel computation of kernels, highlighting the relationship between kernel size and circuit complexity across various problems.

Findings

Exponential kernels for vertex cover are computable by AC$^0$-circuits.

Quadratic kernels for vertex cover are computable by TC$^0$-circuits.

Linear kernels for vertex cover require randomized NC-circuits, with derandomization linked to matching problem complexity.

Abstract

Parallel fixed-parameter tractability studies how parameterized problems can be solved in parallel. A surprisingly large number of parameterized problems admit a high level of parallelization, but this does not mean that we can also efficiently compute small problem kernels in parallel: known kernelization algorithms are typically highly sequential. In the present paper, we establish a number of upper and lower bounds concerning the sizes of kernels that can be computed in parallel. An intriguing finding is that there are complex trade-offs between kernel size and the depth of the circuits needed to compute them: For the vertex cover problem, an exponential kernel can be computed by AC$^0$-circuits, a quadratic kernel by TC$^0$-circuits, and a linear kernel by randomized NC-circuits with derandomization being possible only if it is also possible for the matching problem. Other natural problems for which similar (but quantitatively different) effects can be observed include tree decomposition problems parameterized by the vertex cover number, the undirected feedback vertex set problem, the matching problem, or the point line cover problem. We also present natural problems for which computing kernels is inherently sequential.