Stable Matchings in Practice: A Constraint Programming Approach
Zhaohong Sun, Naoyuki Yamada, Yoshihiro Takenami, Daisuke Moriwaki, Makoto Yokoo
TL;DR
This paper tackles the waiting-child problem in Japan's daycare market, a two-sided matching setting augmented by transfers, siblings, and transferable quotas, formalized as $I=(C,F,D,\omega,\succ_F,\succ_C,\succ_D,G,Q)$. It introduces a constraint programming–based approach (CP-SAT via Google OR-Tools) to compute a stable, individually rational, and feasible matching that minimizes blocking coalitions and maximizes the number of children matched, using variables such as $x[f,p]$, $x[c,d]$, $\alpha[f,p]$, $\gamma[f,p,d,G_d]$, and $\beta[f,p]$, with $∑$-style objective formulations $∑_{f,p} β[f,p]$ and $∑_{c} ∑_{p} x[c,p]$. The authors establish NP-hardness results for stability-involving daycare matchings, provide a practical CP model, and demonstrate substantial empirical gains over current commercial methods on real municipal data (Shibuya, Tama, Koriyama), including improvements in both matches and stability, with data sizes ranging from 550 to 1589 children and 33 to 86 daycares. The work also highlights indifference handling, limited blocking coalitions, and transferable quotas as factors affecting outcomes, and discusses broader applicability to hospital-doctor and school-choice problems, along with public policy implications and open-source release.
Abstract
We study a practical two-sided matching problem of allocating children to daycare centers, which has significant social implications. We are cooperating with several municipalities in Japan and our goal is to devise a reliable and trustworthy clearing algorithm to deal with the problem. In this paper, we describe the design of our new algorithm that minimizes the number of unmatched children while ensuring stability. We evaluate our algorithm using real-life data sets, and experimental results demonstrate that our algorithm surpasses the commercial software that currently dominates the market in terms of both the number of matched children and the number of blocking coalitions (measuring stability). Our findings have been reported to local governments, and some are considering adopting our proposed algorithm in the near future, instead of the existing solution. Moreover, our model and algorithm have broader applicability to other important matching markets, such as hospital-doctor matching with couples and school choice with siblings.