arXiv subjects

Yuhui Zhai

Publications and source records attributed to Yuhui Zhai.

4 recordsLinked to original sources

arXiv2026

Deterministic and Random Bipartite Matching on General Networks: Convex Flow Reformulation, Asymptotic Properties, and Fast Algorithms

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.

math.OC↗

arXiv2026

Developing Fundamental Diagrams for Urban Air Mobility Traffic Based on Physical Experiments

Urban Air Mobility (UAM) is an emerging application of unmanned aerial vehicles that promises to reduce travel time and alleviate congestion in urban transportation systems. As drone density increases, UAM traffic is expected to experience congestion similar to that in ground traffic. However, the fundamental characteristics of UAM traffic, particularly under real-world operating conditions, remain largely unexplored. This study proposes a general framework for constructing the fundamental diagram (FD) of UAM traffic by integrating theoretical analysis with physical experiments. To the best of our knowledge, this is the first study to derive UAM FDs using real-world physical experiment data. On the theoretical side, we design two drone control laws for collision avoidance and develop simulation-based traffic generation methods to produce diverse UAM traffic scenarios. Based on Edie's definition, traffic flow theory is then applied with a near-stationary traffic condition filtering method to construct the FD. To account for real-world disturbances and modeling uncertainties, we further conduct physical experiments on a reduced-scale testbed using Bitcraze Crazyflie drones. Both simulation and physical experiment trajectory data are collected and organized into the UAMTra2Flow dataset, which is analyzed using the proposed framework. Preliminary results indicate that classical FD structures for ground transportation, especially the Underwood model, are applicable to UAM systems. Notably, FD curves obtained from physical experiments exhibit deviations from simulation-based results, highlighting the importance of experimental validation. Finally, results from the reduced-scale testbed are scaled to realistic operating conditions to provide practical insights for future UAM traffic systems. The dataset and code for this paper are publicly available at https://github.com/CATS-Lab/UAM-FD.

cs.RO↗

arXiv2025

Expected Optimal Distances of Random Bipartite Matching in $D$-dimensional Spaces: Approximate Formulas and Applications to Mobility Services

Although many well-known algorithms can solve each bipartite matching problem instance efficiently, it remains an open question how one could estimate the expected optimal matching distance for arbitrary numbers of randomly distributed vertices in $D$-dimensional spaces (referred to as a random bipartite matching problem, or RBMP). This paper proposes a comprehensive modeling framework that yields closed-form approximate formulas for estimating the expected optimal matching cost across three interrelated but increasingly complex versions of RBMPs: (i) RBMP-I, where edge costs are independently and identically distributed (i.i.d.); (ii) RBMP-S, where edge costs represent distances between vertices uniformly distributed on the surface of a hyper-sphere in a $D$-dimensional Euclidean space; and (iii) RBMP-B, where the vertices are uniformly distributed in a hyper-ball within a $D$-dimensional L$^p$ metric space. A series of Monte-Carlo simulation experiments are conducted to verify the accuracy of the proposed formulas under varying parameter combinations. These proposed distance estimates could be key for strategic performance evaluation and resource planning in a wide variety of application contexts. As an illustration, we focus on on-demand mobility services (e.g., e-hailing taxi system). We show how the proposed distance formulas provide a theoretical foundation for the empirically assumed Cobb-Douglas matching function in the field, and reveal conditions under which it can work well. Our formulas can also be easily incorporated into optimization models to select on-demand mobility operation strategies (e.g., whether newly arriving customers shall be instantly matched or pooled into a batch for matching). Agent-based simulations are conducted to verify the predicted performance of the demand pooling strategy for two types of e-hailing taxi systems.

math.OC↗

arXiv2025

Average Distance of Random Bipartite Matching in One-dimensional Space and Networks

The bipartite matching problem is widely applied in the field of transportation; e.g., to find optimal matches between supply and demand over time and space. Recent efforts have been made on developing analytical formulas to estimate the expected matching distance in bipartite matching with randomly distributed vertices in two- or higher-dimensional spaces, but no accurate formulas currently exist for one-dimensional problems. This paper presents a set of closed-form formulas, without curve-fitting, that can provide accurate average distance estimates for one-dimensional random bipartite matching problems (RBMP). We first focus on a lattice case and propose a new method that relates the corresponding matching distance to the area size between a random walk path and the x-axis. This result directly leads to a straightforward closed-form formula for balanced RBMPs. For unbalanced RBMPs on a lattice, we first analyze the properties of an unbalanced random walk that can be related to balanced RBPMs after optimally removing a subset of unmatched points, and then derive a set of approximate formulas. Additionally, we build upon an optimal point removal strategy to derive a set of recursive formulas that can provide more accurate estimates. Then, we extend the results to three problem variants, including RBMPs with periodic boundaries, uniformly distributed points, and arbitrary-length line. Last, we shift our focus to regular networks, and use the one-dimensional results as building blocks to derive RBMP formulas. To verify the accuracy of the proposed formulas, a set of Monte-Carlo simulations are generated for a variety of matching problems settings. Results indicate that our proposed formulas provide quite accurate distance estimations for one-dimensional line segments and networks under a variety of conditions.

math.OC↗