Davide Cherubini - PhD Thesis - UniCA Eprints

6.2 Graphs and Network Flows6.2.2 The Maximum Flow ProblemLet G = (N, A) be a directed graph, u = [u ij ] be a vector, representing the uppercapacity of the arcs, and s and t be two distinct nodes, which are source anddestination of a flow, respectively.The Maximum Flow Problem consists on determining the maximum flow thatcan be sent from source s to destination t.A mathematical formulation of such class of problems is:max v (6.14)∑x js −∑x sj + v = 0 (6.15)(j,s)∈BS(s)∑(j,i)∈BS(i)∑x ji −x jt −(s,j)∈F S(s)∑(i,j)∈F S(i)∑(j,t)∈BS(t) (t,j)∈F S(t)x ij = 0 i ∈ N\{s, t} (6.16)x tj − v = 0 (6.17)0 ≤ x ij ≤ u ij (i, j) ∈ A (6.18)where BS and F S are the Backward and the Froward Star respectively, andv is the flow. Equations 6.15 and 6.17, and 6.16 represent the flow balancingequations at source node s, destination node t, and in a genreic transit node i,respectively.In order to solve this kind of problems many methods have been developedand, the most important are listed in Table 6.1.Algorithm ComplexityFord - Fulkerson [42] O(A · maxflow)Edmonds - Karp [37] O(N · A)Relabel-to-front [30] O(A 3 )Table 6.1: Algorithms to solve the Max Flow ProblemAs aforementioned, the Maximum Flow problem can be seen as a MinimumCost Flow problem if a “virtual” arc (called the return arc) is added to the graphthat goes from the destination t to the source s with a flow value of v.47