Circuits and Backdoors: Five Shades of the SETH

2024-07-12

arXiv.org

AI Summary

The Strong Exponential Time Hypothesis (SETH) faces growing skepticism regarding its plausibility. Method: We propose a systematic framework for weakening SETH by introducing five natural, hierarchically ordered variants grounded in circuit complexity, SAT backtracking gates, graph width parameters, and weighted satisfiability. Contribution/Results: We establish the first five-level equivalence hierarchy among heterogeneous problems—including pathwidth/treewidth/tree-depth modulators, strong 2-SAT/Horn backtracking gates, and shallow-circuit SAT—revealing their fine-grained computational equivalence. Using parameterized reductions, circuit modeling, and graph-structural modularization, we prove that breaking brute-force algorithms for diverse classical problems is computationally equivalent under these assumptions. This unifies explanations of SETH-dependent phenomena and provides a more credible, layer-wise verifiable foundation for fine-grained conditional lower bounds.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The SETH is a hypothesis of fundamental importance to (fine-grained) parameterized complexity theory and many important tight lower bounds are based on it. This situation is somewhat problematic, because the validity of the SETH is not universally believed and because in some senses the SETH seems to be"too strong"a hypothesis for the considered lower bounds. Motivated by this, we consider a number of reasonable weakenings of the SETH that render it more plausible, with sources ranging from circuit complexity, to backdoors for SAT-solving, to graph width parameters, to weighted satisfiability problems. Despite the diversity of the different formulations, we are able to uncover several non-obvious connections using tools from classical complexity theory. This leads us to a hierarchy of five main equivalence classes of hypotheses, with some of the highlights being the following: We show that beating brute force search for SAT parameterized by a modulator to a graph of bounded pathwidth, or bounded treewidth, or logarithmic tree-depth, is actually the same question, and is in fact equivalent to beating brute force for circuits of depth $epsilon n$; we show that beating brute force search for a strong 2-SAT backdoor is equivalent to beating brute force search for a modulator to logarithmic pathwidth; we show that beting brute force search for a strong Horn backdoor is equivalent to beating brute force search for arbitrary circuit SAT.

Problem

• Weaken SETH through circuit complexity and backdoors
• Classify SETH-based lower bounds into equivalence hierarchies
• Characterize parameterized Independent Set complexity for graph classes

Innovation

• Weakened SETH hypotheses from circuit complexity
• Hierarchy of five equivalence classes for complexity
• Parameterized Independent Set algorithms for graph classes