NP membership for general quadratic word equations

Determine whether satisfiability for general quadratic word equations, in which each variable occurs at most twice in total, belongs to the complexity class NP.

Background

A quadratic word equation restricts every variable to at most two total occurrences across both sides of the equation. The paper proves NP-completeness for the intermediate class of quadratic word equations with a linear side, where every variable occurs at most once on one designated side. This leaves unresolved the broader class of arbitrary quadratic word equations: NP-hardness is known, and satisfiability is decidable within PSPACE, but no NP-membership result is established. Resolving this problem would determine whether the polynomial-certificate phenomenon proved for the intermediate class extends to all quadratic word equations.

References

For general quadratic word equations, in which each variable occurs at most twice in total, satisfiability is NP-hard, but its membership in NP remains open.

— Quadratic Word Equations with a Linear Side: Polynomial Nielsen Graph Diameter and NP-Completeness  (2609.21785 - Yonemoto, 18 Sep 2026) in Abstract; see also Section 1, paragraph “Occurrence restrictions and known bounds”