Abstract
Let G = G(A∪B,A×B), with |A| = |B| = n, be a weighted bipartite graph, and let d(·,·) be the cost function on the edges. Let w(M) denote the weight of a matching in G, and M* a minimum-cost perfect matching in G. We call a perfect matching M c-approximate, for c ≥ 1, if w(M) ≤ c · w(M*). We present three approximation algorithms for computing minimum-cost perfect matchings in G.
First, we consider the case when d(·,·) is a metric. For any δ > 0, we present an algorithm that, in O(n2+δ log n log2(1/δ)) time, computes a O(1/δα)-approximate matching of G, where α = log3 2 ≈ 0.631. Next, we assume the existence of a dynamic data structure for answering approximate nearest neighbor (ANN) queries under d(··). Given two parameters ε, &delta ∈ (0, 1), we present an algorithm that, in O(ε---2n1+δτ (n, ε) log2(n/ε) log(1/δ)) time, computes a O(1/δα)-approximate matching of G, where α = 1 + log2(1 + ε) and τ (n; ε) is the query and update time of an (ε/2)-ANN data structure.
Finally, we present an algorithm that works even if d(·,·) is not a metric but admits an ANN data structure for d(·,·). In particular, we present an algorithm that computes, in O(ε---1n3/2τ (n, ε) log4(n/ε) log Δ) time, a (1 + ε)-approximate matching of A and B; here Δ is the ratio of the largest to the smallest-cost edge in G, and τ (n, ε) is the query and update time of an (ε/c)-ANN data structure for some constant c > 1.
We show that our results lead to faster matching algorithms for many geometric settings.
