# L14-minmax - Totally Unimodular Matrices Combinatorial...

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Totally Unimodular Matrices Lecture 11: Feb 23 Combinatorial Algorithm Min-Max Theorem 0 -1 1 1 0 -1 -1 1 0

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2 Player Game 2 Player Game 0 -1 1 1 0 -1 -1 1 0 Row player Column player Row player tries to maximize the payoff, column player tries to minimize Strategy: A probability distribution
2 Player Game 2 Player Game A(i,j) Row player Column player Strategy: A probability distribution You have to decide your strategy first. Is it fair??

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Von Neumann Minimax Theorem Von Neumann Minimax Theorem Strategy set Which player decides first doesn’t matter! e.g. paper, scissor, rock.
Key Observation Key Observation If the row player fixes his strategy, then we can assume that y chooses a pure strategy Vertex solution is of the form (0,0,…,1,…0), i.e. a pure strategy

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Key Observation Key Observation similarly
Primal Dual Programs Primal Dual Programs duality

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Other Applications Other Applications Analysis of randomized algorithms (Yao’s principle) Cost sharing Price setting
Totally Unimodular Matrices Vertex solution: unique solution of n linearly independent tight inequalities m constraints, n variables Can be rewritten as: That is:

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Totally Unimodular Matrices When does has an integral solution x? Assuming all entries of A and b are integral By Cramer’s rule where A i is the matrix with each column is equal to the corresponding column in A except the i-th column is equal to b. x would be integral if det(A) is equal to +1 or -1.
Totally Unimodular Matrices A matrix is totally unimodular if the determinant of each square submatrix of is 0, -1, or +1.

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L14-minmax - Totally Unimodular Matrices Combinatorial...

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