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## CSC5160: Combinatorial Optimization and Approximation Algorithms Topic: Min-Max Theorem Date: 29/02/2008

### Abstract

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. 14.1 Minimax Theorem 14.1.1 Two Player Game Consider the two player game “Rock-Scissors-Paper”, where two persons simultaneously make a hand sign corresponding to one of the three items. Playing “Rock”(R) beats “Scissors”(S), “Scissors” beats “Paper”(P), and “Paper ” beats “Rock”. When both persons play the same action(both R, both S, or both P), then it is a draw game. We assume the payoff for a win is 1, the payoff for losing is-1 and the payoff is 0 for a draw. For two players, we have the following payoff table. In game theory, this is the two player zero-sum game and based on the definition of Nash equilibrium, we know there is no pure strategy solution for this game and the optimal mixed solution is (1/3,1/3,1/3). We will formulate this game as a linear programing problem and try to solve it based on the duality theory

Topics: Theorem 14.1.1 (Von Neumann Minimax Theorem) For every two-person, zero-sum game
Year: 2008
OAI identifier: oai:CiteSeerX.psu:10.1.1.134.216
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