Parallel Complexity of Geometric Bipartite Matching

In this work, we study the parallel complexity of the geometric minimum-weight bipartite perfect matching (GWBPM) problem in ℝ². Here our graph is the complete bipartite graph G on two sets of points A and B in ℝ² (|A| = |B| = n) and the weight of each edge (a, b) ∈ A × B is the ℓ_p distance (for some integer p ≥ 2) between the corresponding points, i.e., ||a - b||_p. The objective is to find a minimum weight perfect matching of A ∪ B. In their seminal work, Mulmuley, Vazirani, and Vazirani (STOC 1987) showed that the weighted perfect matching problem on general bipartite graphs is in RNC. Almost three decades later, Fenner, Gurjar, and Thierauf (STOC 2016) showed that the problem is in Quasi-NC. Both of these results work only when the weights are of O(log n) bits. It is a long-standing open question to show the problem to be in NC. First, we show that in a geometric bipartite graph under the ℓ_p metric for any p ≥ 2, unless we take Ω(n) bits of approximation for weights, we cannot distinguish the minimum-weight perfect matching from other perfect matchings. This means that we cannot hope for an MVV-like NC/RNC algorithm for solving GWBPM exactly (even when vertex coordinates are small integers). Next, we give an NC algorithm (assuming vertex coordinates are small integers) that solves GWBPM up to 1/poly(n) additive error, under the lp metric for any p ≥ 2.