Csc5160: Combinatorial Optimization and Approximation Algorithms Topic: Min-max Theorem

In this lecture, we show the applications of the strong duality theorem, and discuss how to obtain min-max theorems and combinatorial algorithms from linear programming. We first introduce the 2 player, zero-sum game and show that this can be solved by minimax theorem and we also prove the minimax theorem by the LP-duality theorem. After that, we introduce some applications of minimax theorem, such as, analysis of randomized algorithm (Yao’s principle), cost sharing and price setting. Then we show the definition of totally unimodular matrices and prove the important theorem of totally unimodular matrices that if A is totally unimodular, then every vertex solution of Ax > b is integral. Based on this theorem, we show that the min-max theorems for bipartite matching and the maximum flow problem can be obtained by the strong duality theorem. At the end, simplex method will be discussed and we will summarize the polynomial time solvable combinatorial optimization problems and see the role of linear programming in combinatorial optimization.