Faster Algorithms for the Geometric Transportation Problem

Let R,B ⊂ R, for constant d, be two point sets with |R| + |B| = n, and let λ : R ∪ B → N such that ∑ r∈R λ(r) = ∑ b∈B λ(b) be demand functions over R and B. Let d(·, ·) be a suitable distance function such as the Lp distance. The transportation problem asks to find a map τ : R × B → N such that ∑ b∈B τ(r, b) = λ(r), ∑ r∈R τ(r, b) = λ(b), and ∑ r∈R,b∈B τ(r, b)d(r, b) is minimized. We present three new results for the transportation problem when d(·, ·) is any Lp metric: For any constant > 0, an O(n1+ ) expected time randomized algorithm that returns a transportation map with expected cost O(log2(1/ )) times the optimal cost. For any > 0, a (1 + )-approximation in O(n3/2 −d polylog(U) polylog(n)) time, where U = maxp∈R∪B λ(p). An exact strongly polynomial O(n2 polylogn) time algorithm, for d = 2. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems