Bounded-occurrence complexity of LocalLexMax-4-SAT

Determine whether LocalLexMax-4-SAT remains PLS-hard when every variable occurs at most a fixed constant number of times, in order to establish corresponding bounded-family-size hardness for non-wasteful matchings that are fair by master list and resettlement.

Background

The paper proves that finding a feasible and individually rational matching that is non-wasteful and fair by master list and resettlement is PLS-complete. Its reduction from LocalLexMax-4-SAT yields a bound on family size in terms of the maximum number of occurrences of any variable, denoted by Delta.

A constant bound on family size would follow if LocalLexMax-4-SAT were known to be PLS-hard for formulas with bounded variable occurrence. The paper explicitly states that this bounded-occurrence complexity question is unresolved, so it limits the corresponding bounded-family-size strengthening of the matching hardness result.

References

The qualification in Corollary~\ref{cor:fmr:bounded-transfer-consistent} is necessary because the complexity of bounded-occurrence LocalLexMax-4-SAT is currently left open~\citep{ScTa25a}.

— Fairness and Non-wastefulness in Matching with Siblings and Initial Enrollments: Compatibility and Complexity  (2610.01041 - Aziz et al., 1 Oct 2026) in Corollary 1, paragraph immediately following the corollary in Section 'Computational Complexity of Problems Related to FMR'

Corollary~\ref{cor:FN:SFMR} is an existence result. It does not imply a polynomial-time algorithm for the SFMR-based variant of FN, since verifying SFMR is coNP-complete and the existence algorithm for an SFMR matching is not known to run in polynomial time.

— Fairness and Non-wastefulness in Matching with Siblings and Initial Enrollments: Compatibility and Complexity  (2610.01041 - Aziz et al., 1 Oct 2026) in Paragraph immediately following Corollary 2 in Section 'Fairness-preserving Non-wastefulness'