Bike rebalancing: How to find a balanced matching in the k center problem?

In the bike-sharing system, inspired by the problem proposed by O’Mahony and Shmoys (AAAI 2015), we present a new model called the balanced k center problem with matching constraints. Given a network containing the same number of two types of points, supply points and demand points, we aim to allocate k (supply) centers for supply points, and k (demand) centers for demand points. The k supply centers should be perfectly matched with the k demand centers, within an allowed distance. The goal is to minimize the maximum distance from each supply (or demand) point to its closest supply (or demand) centers.The model is motivated by the real-world application in rebalancing bikes, where the bikes are collected in supply centers, and delivered to the matched demand centers, and then distributed from there to serve their demand vertices. Therefore, to successfully complete this rebalancing procedure, two matched groups must have the same number of points. Otherwise, the bikes collected at the supply center may not be enough to serve the demand stations assigned to the demand center.To fully capture the problem, we bring an identical restriction L, which restricts the exact number of vertices assigned to a single center. As a result, all matched centers must serve the same number of bikes. Hence, they are balanced. Our main result is a 9 approximation algorithm. The algorithm is built on the technique called transfer preserving network that successfully combines the approach for the capacitated k center problem with the maximum flow network.