Davide Cherubini - PhD Thesis - UniCA Eprints

6.3 Multicommodity Flow Problemscommodity has its own single or multiple sources s i and its own single or multipledestinations t i .In the integer multicommodity flow problem, the capacities and flows are restrictedto be integers. Unlike the single commodity flow problem, for problemswith integral capacities and demands, the existence of a feasible fractional solutionto the multicommodity flow problem does not guarantee a feasible integralsolution.In typical telecommunication systems model an extra constraint, which maybe imposed, is to restrict each commodity to be sent along a single path or to besplit in equal parts and sent along multiple equivalent paths as seen in chapter 2.Moreover, adding a further constraint representing the survivability of the networkin case of failure makes the model NP-hard.In that case, a new formulation is needed, and the present work is focusedon the development of Multicommodity Min-Cost Flow models that match theproblem statement introduced in chapter 3 and that reduce the complexity.6.3.1 Multicommodity Min-Cost FlowLet G = (N, A) be a directed graph with n nodes and m arcs, and x = [x 1 , · · · , x k ]be a vector, representing the multicommodity flow, of k distinct flow vectors [45].A linear Multicommodity Min-Cost Flow problem can be formalized as anextension of the single commodity formulation as follows:min ∑ h∈K∑x h ij −∑j:(i,j)∈A j:(j,i)∈A∑c h ijx h ij (6.24)(i,j)∈Ax h ji = b h i ∀i ∈ N, ∀h ∈ K (6.25)0 ≤ x h ij ≤ u h ij ∀(i, j) ∈ A, ∀h ∈ K (6.26)∑x h ij ≤ u ij ∀(i, j) ∈ A (6.27)h∈Kwith K = {1, · · · , k}.This representation is called node-arc formulation and describes an LP problem;equation 6.24 states that the k commodities have to be “routed” over the50