AI 中文总结
研究有限平面点集的双色最大和匹配及匹配平衡，证明最大和匹配与有向红圈增益的关系并得出最优性条件，刻画平衡匹配，揭示平衡与多种条件等价，在不同情形下有不同分类。
AI 中文摘要
我们研究有限平面点集的最大和红-蓝匹配及匹配平衡。对于红-蓝完美匹配\(M = \{(a_i,b_i): 1 \le i \le n\}\)，定义有向红圈的增益为循环移动相应蓝伙伴时总权重的变化。证明\(M\)是最大和匹配当且仅当每个有向红圈的增益非正，并从距离差区域的循环相交得出最优性的几何充分条件。接着刻画平衡匹配，其中所有红-蓝完美匹配具有相同总权重。平衡被证明等同于圈增益消失、距离矩阵的加法形式以及距离差函数的公共水平集条件。在平方欧几里得情形下产生正交分类，在欧几里得情形下产生双曲水平集描述和非退化设置下的共线分离分类。
英文摘要
We study maximum-sum red-blue matchings and matching equilibrium for finite planar point sets. For a red-blue perfect matching $M = \{(a_i,b_i) : 1 \le i \le n\}$, we define the gain of a directed red cycle as the change in total weight produced by cyclically shifting the corresponding blue partners. We prove that $M$ is maximum-sum if and only if every directed red cycle has nonpositive gain, and we derive a geometric sufficient condition for optimality from cyclic intersections of distance-difference regions. We then characterize balanced matchings, in which all red-blue perfect matchings have the same total weight. Equilibrium is shown to be equivalent to vanishing cycle gains, to an additive form of the distance matrix, and to a common level-set condition for distance-difference functions. In the squared Euclidean case this yields an orthogonality classification, while in the Euclidean case it yields a hyperbolic level-set description and a collinear-separation classification in the nondegenerate setting.
Comments17 pages, 6 figures
