Lecture notes for “ Analysis of Algorithms ” : Minimum cost flow and weighted bipartite matching

We present strongly polynomial time algorithms for the minimum cost flow problem and for the weighted bipartite matching problem, also known as the assignment problem. 1 The minimum cost flow problem A flow network with prices, demands and capacities N = (G, a, b, c) is composed of a directed graph G = (V, E), a cost function a : E → R, a demand function b : V → R, and a capacity function c : E → R. A feasible flow f in N is a function f : E → R that satisfies that satisfies the following two conditions: 0 ≤ f(e) ≤ c(e) , e ∈ E (1) f(in(v))− f(out(v)) = b(v) , v ∈ V (2) The cost of the flow f is: Cost(f) = ∑