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

TL;DR

This paper develops approximate formulas for the expected optimal matching distance in random bipartite matching problems across various $D$-dimensional spaces, with applications to mobility services and resource planning.

Contribution

It introduces a comprehensive modeling framework with closed-form formulas for estimating matching costs in different $D$-dimensional distributions, verified by simulations.

Findings

Formulas accurately estimate expected matching distances across scenarios.

Distance estimates support theoretical foundations of matching functions in mobility.

Formulas enable optimization of matching strategies in on-demand mobility services.

Abstract

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.