Minimax Programs, Bitonic Columns and Pq Trees

1 Overview Among hard combinatorial optimization problems there is an increasing body of special cases known in which optimal solutions can be obtained eeciently. These special cases may be expressed as constraints on graph structure, such as chordal or interval graphs 4, 6] which allow linear-time solution of problems like vertex coloring, or Monge matrix edge weights which allow eecient solution of TSP 2], or as constraints on a general integer programming formulation 8]. These cases are of interest because they sometimes arise in natural problems we wish to solve, techniques for their solution can lead to more eecient (perhaps approximate) algorithms for other problems, and their study can lead to greater insight into what makes some problem instances hard. We introduce a new combinatorial optimization problem with similarity to a linear program and the set cover problem. A minimax program is formulated just like a linear program except that the addition operator is replaced by the maximum operator in the constraint