AI 中文总结
本文否证了Aigner关于星森林搜索的majorization猜想，通过六次测试反例及解析反例族表明该条件不充分，并证明五以内成立。
AI 中文摘要
在恰好有两个缺陷物品的自适应定量群组测试中，每次测试报告所选子集中存在多少个缺陷物品。我们研究可能的缺陷对构成星森林的边的配置。Aigner证明了在规定的测试次数内识别缺陷对所需的关于有序星大小的一个必要majorization条件，并猜想该条件是充分的。我们通过一个明确的六次测试反例否证了这一猜想，并为每个测试预算$k\ge6$给出了一个解析的反例族。该障碍利用两个紧的前缀和来迫使第一次测试的结果产生不兼容的要求。我们还通过穷举计算结合解析化简证明了对于$0\le k\le5$，逆命题成立。因此，六是majorization单独失效的第一个测试预算。这些反例及其无限扩展不依赖于穷举计算。
英文摘要
In adaptive quantitative group testing with exactly two defective items, each test reports how many defectives lie in a chosen subset. We study configurations in which the possible defective pairs form the edges of a star forest. Aigner proved a necessary majorization condition on the ordered star sizes for identifying the defective pair within a prescribed number of tests, and conjectured that this condition was sufficient. We disprove the conjecture by an explicit six-test counterexample and give an analytic family of counterexamples for every test budget $k\ge6$. The obstruction uses two tight prefix sums to force incompatible demands on the outcomes of the first test. We also prove, by exhaustive computation combined with analytic reductions, that the converse holds for $0\le k\le5$. Thus six is the first test budget at which majorization alone fails. The counterexamples and their infinite extension do not depend on the exhaustive computation.
Comments12 pages
