Efficient Algorithms for Geometric Partial Matching
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Let A and B be two point sets in the plane of sizes r and n respectively (assume r ≤ n), and let k be a parameter. A matching between A and B is a family of pairs in A × B so that any point of A ∪ B appears in at most one pair. Given two positive integers p and q, we define the cost of matching M to be c(M) = ∑_(a, b) ∈ Ma-b_p^q where ·_p is the L_p-norm. The geometric partial matching problem asks to find the minimum-cost size-k matching between A and B. We present efficient algorithms for geometric partial matching problem that work for any powers of L_p-norm matching objective: An exact algorithm that runs in O((n + k^2) polylog n) time, and a (1 + ε)-approximation algorithm that runs in O((n + k√(k)) polylog n ·ε^-1) time. Both algorithms are based on the primal-dual flow augmentation scheme; the main improvements involve using dynamic data structures to achieve efficient flow augmentations. With similar techniques, we give an exact algorithm for the planar transportation problem running in O({n^2, rn^3/2}polylog n) time.
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