Aigner's majorization conjecture in quantitative group testing: counterexamples and eventual sufficiency for star forests

Abstract

We refute Aigner's majorization conjecture for star forests in adaptive quantitative group testing with two defective items. Each test counts the defectives in a chosen subset; the possible defective pairs form the edges of a star forest. Aigner's necessary majorization condition is sufficient through five tests but is not sufficient at any budget from six onward. Nevertheless, for every fixed number of stars, it becomes sufficient once the test budget is large enough. We prove that majorization is sufficient for all forests with at most twenty-nine stars, and that this bound is sharp. We also construct counterexamples that pass arbitrarily many successive compatibility checks based on majorization. The five-test converse, counterexamples, eventual sufficiency theorem and unbounded-delay construction have analytic proofs. The sharp finite support bounds also use exact integer-inequality certificates.