On Kernelization with Access to NP-Oracles

2025-08-14

AI Summary

This work addresses the limited preprocessing capability for high-level polynomial-hierarchy problems—e.g., Σ₂^P-complete problems—where classical kernelization fails. Method: We introduce and formalize the *P^NP-kernel*, a novel kernelization framework permitting polynomial-time access to an NP oracle (e.g., a SAT solver). Our approach integrates parameterized complexity analysis, lower-bound techniques, and meta-theorems, and implements concrete algorithms using SAT and ILP solvers. Contribution/Results: We establish tight existence and impossibility bounds for P^NP-kernels; obtain the first positive and negative results for graph problems (e.g., variants of Dominating Set) and discovery-type problems; and provide the first preprocessing paradigm for higher-order computational problems that balances expressive power with analytical tractability—thereby extending the scope of classical kernelization beyond the NP level.

Technology Category

• Machine Learning: Kernel Methods
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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].

Problem

• Extends kernelization to use SAT-solvers for preprocessing
• Introduces P^NP-Kernel for higher complexity problems
• Provides theoretical framework for kernelization with NP-oracles

Innovation

• Introduces P^NP-Kernel using SAT-solvers
• Relaxes kernelization to allow NP-oracle calls
• Proves positive and negative results framework