Hard Clique Formulas for Resolution

2026-01-18

AI Summary

This work resolves a long-standing open problem regarding the unconditional refutation complexity of the k-clique problem in the Resolution proof system. By explicitly reducing sparse unsatisfiable 3-CNF formulas—known to require exponential-size Resolution refutations—to instances of the k-clique problem and analyzing their natural CNF encodings, the authors construct explicit k-clique instances that necessitate n^{Ω(k)} steps to refute in Resolution. This lower bound is unconditional, relying on no complexity-theoretic assumptions, thereby establishing the first such hardness result for the canonical CNF encoding of k-clique in Resolution. Moreover, the result provides a proof-complexity-based reaffirmation of the classical conditional hardness of k-clique under the Exponential Time Hypothesis (ETH), now grounded in an unconditional setting.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• Security and Privacy: Data transparency and provenance

Abstract

We show how to convert any unsatisfiable 3-CNF formula which is sparse and exponentially hard to refute in Resolution into a negative instance of the $k$-clique problem whose corresponding natural encoding as a CNF formula is $n^{\Omega(k)}$-hard to refute in Resolution. This applies to any function $k = k(n)$ of the number $n$ of vertices, provided $k_0 \leq k \leq n^{1/c_0}$, where $k_0$ and $c_0$ are small constants. We establish this by demonstrating that Resolution can simulate the correctness proof of a particular kind of reduction from 3-SAT to the parameterized clique problem. This also re-establishes the known conditional hardness result for $k$-clique which states that if the Exponential Time Hypothesis (ETH) holds, then the $k$-clique problem cannot be solved in time $n^{o(k)}$. Since it is known that the analogue of ETH holds for Resolution, unconditionally and with explicit hard instances, this gives a way to obtain explicit instances of $k$-clique that are unconditionally $n^{\Omega(k)}$-hard to refute in Resolution. This solves an open problem that appeared published in the literature at least twice.

Problem

• Hard Clique Formulas
• Resolution
• 3-CNF
• k-clique
• Proof Complexity

Innovation

• Resolution
• k-clique
• proof complexity
• hardness amplification
• 3-SAT