A Polynomial Combinatorial Algorithm for Generalized Minimum Cost Flow

We propose the first combinatorial solution to one of the most classic problems in combinatorial optimization: the generalized minimum cost flow problem (flow with losses and gains). Despite a rich history dating back to Kantorovich and Dantzig, until now, the only known way to solve the problem in polynomial-time was via general purpose linear programming techniques. Polynomial combinatorial algorithms were previously known only for the version of our problem without costs. We design the first such algorithms for the version with costs. Our algorithms also find provably good solutions faster than optimal ones, providing the first strongly polynomial approximation schemes for the problem. Our techniques extend to optimize linear programs with two variables per inequality. Polynomial combinatorial algorithms were previously developed for testing the feasibility of such linear programs. Until now, no such methods were known for the optimization version.

• Type: Conference
• Year: 1999
• Where: STOC
• Authors: Kevin D. Wayne
• Keywords: Algorithms, Cost flow Problem, Polynomial Combinatorial Algorithms, STOC 1999, first Combinatorial Solution