Combinatorial Approximation Algorithms for Generalized Flow Problems

Generalized network flow problems generalize normal network flow problems by specifying a flow multiplier (a) for each arc a. For every unit of flow entering the arc, (a) units of flow exit. Flow multipliers permit modelling transforming one type into another and modification of the amount of flow

Combinatorial Approximation Algorithms for Generalized Flow Problems

Generalized network flow problems generalize normal network flow problems by specifying a flow multiplier for each arc. For every unit of flow entering the arc, units of flow exit. Flow multipliers permit modelling transforming one type into another and modification of the amount of flow. For example, currency exchange and water evaporation from canals can be modelled. We present a strongly polynomial algorithm for a single-source generalized shortest paths problem, also called the restricted generalized uncapacitated transshipment problem. We present a left-distributive closed semiring which permits use of the Bellman-Ford algorithm to solve this problem given a guess for the value of the optimal solution. Using Megiddo’s parametric search scheme, we can compute the optimal value in strongly polynomial time. The algorithm’s running time matches the previously best known, but the algorithm is simpler, is based on the well-known theory of closed semirings, and directly works with the given graph. All previous polynomial-time algorithms were based on interior-point methods or directly solved the dual problem and translated the solution back to the primal problem. Using this generalized shortest paths algorithm, we present fully polynomial-time approximation schemes for the generalized versions of the maximum flow, the nonnegative-cost minimum-cost flow, the concurrent flow, the multicommodity maximum-flow, and the multicommodity nonnegative-cost minimum-cost flow problems. For all of these problems except the maximum flow variant, these combinatorial algorithm schemes are the first polynomial-time algorithms not based on interior point methods. All running times are independent of the size of the flow multipliers ’ representation. Also, the generalized ¡concurrent flow and the generalized multicommodity maximum flow approximation schemes are the first known strongly polynomial algorithms.