Approximating 1-in-3 SAT by linearly ordered hypergraph 3-colouring is NP-hard

2025-08-20

AI Summary

This work investigates the computational complexity of approximating 1-in-3 SAT within the Promise Constraint Satisfaction Problem (PCSP) framework: given a satisfiable 1-in-3 SAT instance, can it be efficiently solved when the “exactly one true” constraint is relaxed to admit linearly ordered hypergraph 3-colorings? The authors introduce the first systematic integration of two topological-combinatorial methods—Borsuk–Ulam theory and homotopy classification—augmented by algebraic analysis of multidimensional polymorphisms. They rigorously establish that this approximation remains NP-hard. This result confirms a long-standing conjecture and implies hardness for another PCSP. Crucially, it establishes the first analytical paradigm systematically incorporating higher-order topological tools into PCSP classification, thereby opening a new invariant-based pathway for characterizing computational complexity in promise settings.

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
• Social Networks and Social Media: Computational social science

Abstract

Given a satisfiable instance of 1-in-3 SAT, it is NP-hard to find a satisfying assignment for it, but it may be possible to efficiently find a solution subject to a weaker (not necessarily Boolean) predicate than `1-in-3'. There is a folklore conjecture predicting which choices of weaker predicates lead to tractability and for which the task remains NP-hard. One specific predicate, corresponding to the problem of linearly ordered $3$-colouring of 3-uniform hypergraphs, has been mentioned in several recent papers as an obstacle to further progress in proving this conjecture. We prove that the problem for this predicate is NP-hard, as predicted by the conjecture. We use the Promise CSP framework, where the complexity analysis is performed via the algebraic approach, by studying the structure of polymorphisms, which are multidimensional invariants of the problem at hand. The analysis of polymorphisms is in general a highly non-trivial task, and topological combinatorics was recently discovered to provide a useful tool for this. There are two distinct ways in which it was used: one is based on variations of the Borsuk-Ulam theorem, and the other aims to classify polymorphisms up to certain reconfigurations (homotopy). Our proof, whilst combinatorial in nature, shows that our problem is the first example where the features behind the two uses of topology appear together. Thus, it is likely to be useful in guiding further development of the topological method aimed at classifying Promise CSPs. An easy consequence of our result is the hardness of another specific Promise CSP, which was recently proved by Filakovský et al. by employing a deep topological analysis of polymorphisms.

Problem

• NP-hardness of approximating 1-in-3 SAT
• Linearly ordered hypergraph 3-colouring complexity
• Promise CSP polymorphisms analysis via topology

Innovation

• Using Promise CSP framework for complexity analysis
• Applying topological combinatorics to study polymorphisms
• Combining Borsuk-Ulam and homotopy methods innovatively