On Kernelization with Access to NP-Oracles

TL;DR

This paper introduces P^NP-Kernels, a new framework for kernelization that leverages polynomially many calls to NP-oracles like SAT-solvers, extending the concept to problems beyond the polynomial hierarchy.

Contribution

It formalizes the P^NP-Kernel concept, explores its properties through positive and negative results, and provides a meta-theorem for discovery problems, broadening kernelization theory.

Findings

Established lower bounds for P^NP-Kernels

Proved positive results for certain graph problems

Presented a meta-theorem for discovery problems

Abstract

Kernelization is the standard framework to analyze preprocessing routines mathematically. Here, in terms of efficiency, we demand the preprocessing routine to run in time polynomial in the input size. However, today, various NP-complete problems are already solved very fast in practice; in particular, SAT-solvers and ILP-solvers have become extremely powerful and used frequently. Still, this fails to capture the wide variety of computational problems that lie at higher levels of the polynomial hierarchy. Thus, for such problems, it is natural to relax the definition of kernelization to permit the preprocessing routine to make polynomially many calls to a SAT-solver, rather than run, entirely, in polynomial time. Our conceptual contribution is the introduction of a new notion of a kernel that harnesses the power of SAT-solvers for preprocessing purposes, and which we term a P^NP-Kernel. Technically, we investigate various facets of this notion, by proving both positive and negative results, including a lower-bounds framework to reason about the negative results. Here, we consider both satisfiability and graph problems. Additionally, we present a meta-theorem for so-called "discovery problems". This work falls into a long line of research on extensions of the concept of kernelization, including lossy kernels [Lokshtanov et al., STOC '17], dynamic kernels [Alman et al., ACM TALG '20], counting kernels [Lokshtanov et al., ICTS '24], and streaming kernels [Fafianie and Kratsch, MFCS '14].