A SUBSET-SUM Characterisation of the A-Hierarchy
Jan Gutleben, Arne Meier
TL;DR
The paper addresses the problem of giving a natural, combinatorial characterisation of the A-hierarchy, a parametric analogue of the polynomial hierarchy, by introducing generalised SUBSET-SUM problems that mirror logical alternations. It develops an ARAM-based, tail-nondeterministic framework to characterise A[t] and proves tight completeness results: the ALT_3_SUBSET-SUM variant is in A[3] and A[3]-hard, and the approach extends to ALT_ell_SUBSET-SUM for all levels, with odd ell yielding A[ell]-completeness and even ell yielding CO-ALT_ell completeness. The core contributions are a concrete, natural complete problem family for each level of the A-hierarchy and a transferable method to connect parameterised model checking with arithmetic encodings. This establishes a bridge between predicate-logic model checking and combinatorial SUBSET-SUM constructions, with implications for both theory and potential counting-adapted problems.
Abstract
The A-hierarchy is a parametric analogue of the polynomial hierarchy in the context of paramterised complexity theory. We give a new characterisation of the A-hierarchy in terms of a generalisation of the SUBSET-SUM problem.