Maximum bipartite matching is a fundamental algorithmic problem which can be solved in polynomial time. We consider a natural variant in which there is a separation constraint: the vertices on one side lie on a path or a grid, and two vertices that are close to each other are not allowed to be matched simultaneously. We show that the problem is hard to approximate even for paths, and provide constant-factor approximation algorithms for both paths and grids.

翻译：最大二分图匹配是一个基础算法问题，可在多项式时间内求解。我们考虑一个包含分离约束的自然变体：一侧顶点位于路径或网格上，且相距较近的两个顶点不允许同时匹配。我们证明，即使对于路径情形，该问题也难以近似，并针对路径和网格两种情形提供了常数因子近似算法。