[![Bipartite matching][1]][1]

Suppose a square $[0,1]\times [0,1]$ in which $N$ vehicles $V_i$ and $N$ riders $R_i$ are distributed identically and independently (say, uniform distribution), a bipartite matching (or a permutation, $\pi(i)$) is done between the vehicles and riders with the objective that the total distance 

$$Z=\min_{\pi}\sum_1^N \sqrt{\Vert V_{\pi(i)}-R_i\Vert^2}$$

is minimized.

Since the locations of vehicles and riders are distributed randomly, therefore $Z$ is a random variable. The expectation of $Z$ is thus of interest. The question is how to derive the $E(Z)$.

I have found some related papers, such as

 1. Caracciolo, S., & Sicuro, G. (2015). [Quadratic stochastic Euclidean
    bipartite matching
    problem](https://air.unimi.it/retrieve/handle/2434/342033/488849/Lettera.pdf).
    *Physical review letters*, 115(23), 230601.    
 2. Holroyd, A. E., Pemantle, R., Peres, Y., & Schramm, O. (2009).
    [Poisson
    matching](http://www.numdam.org/article/AIHPB_2009__45_1_266_0.pdf).
    In *Annales de l'Institut Henri Poincaré, Probabilités et
    Statistiques* (Vol. 45, No. 1, pp. 266-287). Institut Henri
    Poincaré. 
 3. Boniolo, E., Caracciolo, S., & Sportiello, A. (2014). [Correlation
    function for the Grid-Poisson Euclidean matching on a line and on a
    circle](https://arxiv.org/pdf/1403.1836.pdf). *Journal of
    Statistical Mechanics: Theory and Experiment*, 2014(11), P11023.

I am trying to read them to figure out how, but their derivation all has some part heavily related to physics and statistical mechanics, which makes me struggle to understand it but I fail.

I was wondering if there is a version with a no-physics-just-operations-research flavor to solve this problem?


  [1]: https://i.sstatic.net/XaC90.png