We characterize a rich class of valuated matroids, called R-minor valuated matroids that includes the indicator functions of matroids, and is closed under operations such as taking minors, duality, and induction by network. We refute the refinement of a 2003 conjecture by Frank, exhibiting valuated matroids that are not R-minor. The family of counterexamples is based on sparse paving matroids. Valuated matroids are inherently related to gross substitute valuations in mathematical economics. By the same token we refute the Matroid Based Valuation Conjecture by Ostrovsky and Paes Leme (Theoretical Economics 2015) asserting that every gross substitute valuation arises from weighted matroid rank functions by repeated applications of merge and endowment operations. Our result also has implications in the context of Lorentzian polynomials: it reveals the limitations of known construction operations.
翻译:我们描述一大批价值丰富的、称为 R-minor valued matilids, 包括了机器人的指数功能,在诸如抓走未成年人、双重性和网络诱导等行动下关闭。我们驳斥了Frank2003年的推测,即展示了非R-minor的有价值类人造机器人的精细图象。反采样的家族以稀疏的铺面类人造假机为基础。有价值类人造假与数学经济学中的粗替代值评估有内在联系。同样,我们驳斥了Ostrovsky和Paes Leme(2015年理论经济学)的“基于机器人的估价预测”这一论调,即通过反复应用合并和捐赠作业,每一种总替代价值都是由加权的类人造假函数产生的。我们的结果还影响到Lorentzian 多边营养学:它揭示了已知建筑作业的局限性。