Strong Structural Bounds for MaxSAT: The Fine Details of Using Neuromorphic and Quantum Hardware Accelerators

2024-12-13

arXiv.org

AI Summary

Efficiently solving MaxSAT on neuromorphic and quantum hardware remains challenging due to limitations of conventional encoding-centric optimization approaches. Method: We propose a structure-aware encoding and reduction theory for MaxSAT, moving beyond mere size minimization. We establish the first linear-time, treewidth-preserving reduction framework from MaxSAT and Max2SAT to QUBO; derive tight complexity lower bounds under ETH/SETH, thereby closing a fundamental gap in incidence treewidth theory; design a primal-treewidth-optimal fixed-parameter QUBO algorithm; and develop an efficient model-counting–based algorithm for tractable MaxSAT fragments. Results: Our work provides both theoretical foundations and practical tools for structured hardware acceleration. The resulting solver achieves time-optimal performance, and empirical evaluation demonstrates significant improvements in real-world solving efficiency.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Hardware-aware ML

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Hardware accelerators like quantum annealers or neuromorphic chips are capable of finding the ground state of a Hamiltonian. A promising route in utilizing these devices is via methods from automated reasoning: The problem at hand is first encoded into MaxSAT; then MaxSAT is reduced to Max2SAT; and finally, Max2SAT is translated into a Hamiltonian. It was observed that different encodings can dramatically affect the efficiency of the hardware accelerators. Yet, previous studies were only concerned with the size of the encodings rather than with syntactic or structural properties. We establish structure-aware reductions between MaxSAT, Max2SAT, and the quadratic unconstrained binary optimization problem (QUBO) that underlies such hardware accelerators. All these problems turn out to be equivalent under linear-time, treewidth-preserving reductions. As a consequence, we obtain tight lower bounds under ETH and SETH for Max2SAT and QUBO, as well as a new time-optimal fixed-parameter algorithm for QUBO. While our results are tight up to a constant additive factor for the primal treewidth, we require a constant multiplicative factor for the incidence treewidth. To close the emerging gap, we supplement our results with novel time-optimal algorithms for fragments of MaxSAT based on model counting.

Problem

• Optimize MaxSAT encoding for hardware accelerators
• Establish structure-aware reductions between MaxSAT and QUBO
• Develop time-optimal algorithms for MaxSAT fragments

Innovation

• Utilizes neuromorphic and quantum hardware accelerators
• Develops structure-aware reductions for MaxSAT problems
• Introduces time-optimal algorithms for QUBO and MaxSAT