Kernelization of Counting Problems

TL;DR

This paper proposes a new framework for preprocessing in parameterized counting problems, focusing on kernelization and polynomial compression, with theoretical bounds and limitations.

Contribution

It introduces a novel framework for counting problem preprocessing, including definitions of compression and kernelization, and provides bounds and limitations for these techniques.

Findings

Framework captures polynomial and lossy kernelization for counting problems.

Defines reduction and lift procedures for problem compression.

Establishes upper and lower bounds for kernelization in counting problems.

Abstract

We introduce a new framework for the analysis of preprocessing routines for parameterized counting problems. Existing frameworks that encapsulate parameterized counting problems permit the usage of exponential (rather than polynomial) time either explicitly or by implicitly reducing the counting problems to enumeration problems. Thus, our framework is the only one in the spirit of classic kernelization (as well as lossy kernelization). Specifically, we define a compression of a counting problem $P$ into a counting problem $Q$ as a pair of polynomial-time procedures: $\mathsf{reduce}$ and $\mathsf{lift}$. Given an instance of $P$, $\mathsf{reduce}$ outputs an instance of $Q$ whose size is bounded by a function $f$ of the parameter, and given the number of solutions to the instance of $Q$, $\mathsf{lift}$ outputs the number of solutions to the instance of $P$. When $P=Q$, compression is termed kernelization, and when $f$ is polynomial, compression is termed polynomial compression. Our technical (and other conceptual) contributions concern both upper bounds and lower bounds.