Faster algorithms for the geometric transportation problem

Let R, B C Rd for constant d, be two point sets with |R| + |B| = n, and let λ: R∪B → ℕ such that Σr∈Rλ(r) = Σb∈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 Σb∈B τ(r, b) = λ(r), Σr∈Rτ(r, b) = λ(b), and σr∈Rb∈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 ϵ > 0, an O(n1+ϵ) expected time randomized algorithm that returns a transportation map with expected cost O(log2(1/ϵ)) times the optimal cost. • For any ϵ > 0, a (1 + ϵ)-approximation in O(n3/2 ϵ-d polylog(U) polylog(n)) time, where U = maxp∈Rcup;B λ (p). •An exact strongly polynomial O(n2 polylogn) time algorithm, for d = 2.

Duke Scholars

Author Debmalya Panigrahi Computer Science