We call a partition of a c-partite tournament into tournaments of order c strong if each tournament is strongly connected.The strong partition number,denoted as ST(r),represents the minimum integer c' such that every regular r-balanced c-partite tournament has a strong partition for all c ≥ c'.Figueroa,Montellano-Ballesteros,and Olsen showed the existence of ST(r)for all r≥2 and proved that 5 ≤ ST(2)≤ 7.In this note,we establish that ST(2)=6 and we also show the unique 2-balanced 5-partite tournament which has no strong partition.
Risk management often plays an important role in decision making un-der uncertainty.In quantitative risk management,assessing and optimizing risk metrics requires efficient computing techniques and reliable theoretical guarantees.In this pa-per,we introduce several topics on quantitative risk management and review some of the recent studies and advancements on the topics.We consider several risk metrics and study decision models that involve the metrics,with a main focus on the related com-puting techniques and theoretical properties.We show that stochastic optimization,as a powerful tool,can be leveraged to effectively address these problems.
针对在时刻表给定条件下如何减少空驶车次和实现具有工作时间公平性的公交车辆调度问题,建立了基于超级时空网络的模型,并设计了一种具有混生、变异和成长三种基本操作的进化求解算法.首先利用超级网络理念,将出场弧、入场弧、接续、实际车次和空驶车次在时空上整合为一个连通的有向超级时空网络.基于超级网络中流量守恒概念,建立了公交车辆调度模型,并通过合理转化将工作时间公平性约束变为具有简单加和特征的目标函数项.利用可行车次覆盖集合的拓扑结构特征,设计了将多个可行解混合后生成新解的混生算子;通过搜索具有回路特征的接续,实现对可行解构成元素的变异操作;通过构建指派网络、计算指派网络中联接的费用,并利用匈牙利算法求解对应指派问题,实现对可行解的成长操作.以上述操作为基础提出了一种新的"3M"进化算法.通过实证分析,验证了模型的合理性与算法的有效性.研究发现:减少空驶车次与平衡车次链之间的实际车次运行时间之间存在相互制约的矛盾,但是与所需的公交车总数不存在必然联系.
Wiener指数作为化学图论中的一个重要的化学指标,是连通图的任意两个顶点的距离之和.Steiner Wiener指数是Wiener指数的一种推广形式,定义为所有k个顶点的集合S的Steiner距离的和,其中S的Steiner距离是包含S的最小连通子图的边数.本文研究了具有最小(大)Steiner Wiener指数的单圈图结构.
(d,w)-控制数是一个度量共享网络资源的可靠性的重要参数.(1,1)-控制数就是图论中的经典控制数,(d,w)-控制数是(1,1)-控制数的一个直接推广.本文给出了计算(1,w)-控制数的一个下界方法和一个上界方法.应用这两个方法,求出了超立方体的(1,n-1)-控制数和(1,n)-控制数,求出了 4基n立方体的(1,2n-1)-控制数和(1,2n)-控制数,求出了n维折叠超立方体的(1,n)-控制数.
