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
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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.
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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
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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
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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.
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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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