一般网络上的确定性与随机二分匹配：凸流重构、渐近性质与快速算法
Deterministic and Random Bipartite Matching on General Networks: Convex Flow Reformulation, Asymptotic Properties, and Fast Algorithms
AI总结：
本文针对一般网络上的二分匹配，提出凸流重构及快速估计算法，证明随机情形下匹配距离的标度律，并数值验证其高效性。
AI中文摘要：
一般网络上的最小距离二分匹配在多个领域有众多应用。本文首先关注确定性问题，并提出一种精确的边可分离凸流重构。通过引入边不平衡分布的平滑单调重排近似，凸流重构可以高效求解。若进一步对凸规划进行基于电阻的一阶近似，则可解析地推导出闭式的一步拉普拉斯估计器。本文还研究了供应点和需求点随机分布的随机问题。我们证明，若供应/需求点分布相同，则期望最优匹配距离随点数的平方根增长，否则随点数线性增长。在前一种情形下，最优流被证明是中心化、对称且次高斯的。在后一种情形下，极限电阻网络刻画了供需不平衡如何重新分布，并启发了一种基于极限电阻近似最优流的快速算法。数值实验表明，所提出的估计器能紧密逼近精确匹配成本，同时大幅减少计算时间。大规模蒙特卡洛模拟数值验证了随机匹配解的已证明理论性质。
英文摘要：
Minimum-distance bipartite matching on general networks has numerous applications various fields. This paper first focuses on deterministic problems and presents an exact edgewise-separable convex-flow reformulation. By introducing a smooth monotone rearrangement approximation of the edge-wise imbalance profiles, the convex-flow reformulation's can be solved efficiently. If we further conduct a first-order resistance-based approximation of the convex program, a one-step Laplacian-based estimator can be analytically derived in closed forms. The paper also studies random problems where supply and demand points are randomly distributed. We show that the expected optimal matching distance scales with the square root of the number of points if the supply/demand point distributions are identical, or linearly otherwise. In the former case, the optimal flow is proven to be centered, symmetric, and sub-Gaussian. In the latter case, the limiting resistance network characterizes how supply-demand imbalance is redistributed and motivates a fast algorithm that approximate the optimal flow based on the limiting resistance. Numerical experiments show that the proposed estimators closely approximate the exact matching cost while substantially reducing computation time. The proven theoretical properties of the random matching solution are numerically verified by large-scale Monte Carlo simulations.
