© Pankaj K. Agarwal, Kyle Fox, Debmalya Panigrahi, Kasturi R. Varadarajan, and Allen Xiao. Let R, B C Rdfor constant d, be two point sets with |R| + |B| = n, and let λ: R∪B → ℕ such that Σr∈Rλ(r) = Σb∈Bλ (b) be demand functions over R and B. Let d(·, ·) be a suitable distance function such as the Lpdistance. The transportation problem asks to find a map τ: R × B → ℕ such that Σb∈Bτ(r, b) = λ(r), Σr∈Rτ(r, b) = λ(b), and σr∈Rb∈Bτ(r, b)d(r, b) is minimized. We present three new results for the transportation problem when d(·, ·) is any Lpmetric: • 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(n3/2ϵ-dpolylog(U) polylog(n)) time, where U = maxp∈Rcup;Bλ (p). •An exact strongly polynomial O(n2polylogn) time algorithm, for d = 2.