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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.

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.

We present several new efficient algorithms for the generalized maximum flow problem. In the traditional maximum flow problem, there is a capacitated network and the goal is to send as much of a single commodity as possible between two distinguished nodes, without exceeding the arc capacity limits. The problem has hundreds of applications including: shipping freight in a transportation network and pumping fluid through a hydraulic network.In traditional networks, there is an implicit assumption that flow is conserved on every arc. Many practical applications violate this conservation assumption.Freight may be damaged or spoil in transit; fluid may leak or evaporate. In generalized networks, each arc has a positive multiplier associated with it, representing the fraction of flow that remains when it is sent along that arc. The generalized maximum flow problem is identical to the traditional maximum flow problem, except that it can also model networks which "leak" flow. Biographical SketchKevin Wayne was born on October 16, 1971 in Philadelphia, Pennsylvania. After failing a placement exam in seventh grade, Kevin was assigned to a remedial math class, where he has found memories of modeling conic sections with clay. By the end of high school, he was taking advanced calculus classes at the University of Pennsylvania, where he realized there was more to math than playing with clay.As an undergraduate at Yale University, Kevin stumbled upon a linear programming course, which introduced him to the area of operations research. In 1993, Yale

Abstract. In the baseball elimination problem, there is a league consisting of n teams. At some point during the season, team i has w i wins and g ij games left to play against team j. A team is eliminated if it cannot possibly finish the season in first place or tied for first place. The goal is to determine exactly which teams are eliminated. The problem is not as easy as many sports writers would have you believe, in part because the answer depends not only on the number of games won and left to play but also on the schedule of remaining games. In the 1960's, Schwartz showed how to determine whether one particular team is eliminated using a maximum flow computation. This paper indicates that the problem is not as difficult as many mathematicians would have you believe. For each team i, let g i denote the number of games remaining. We prove that there exists a value W * such that team i is eliminated if and only if w i + g i < W * . Using this surprising fact, we can determine all eliminated teams in time proportional to a single maximum flow computation in a graph with n nodes; this improves upon the previous best known complexity bound by a factor of n.