Solving Zero-Sum Security Games in Discretized Spatio-Temporal Domains
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چکیده
Proof. we’re following an argument from (Grötschel, Lovász, and Schrijver 1984); proof is provided for completeness. Assume we are given an oracle O that computes LPw. Consider another polyhedron P◦ w = {y : yx ≤ 1,∀x ∈ Pw}. For any y, we can decide whether y ∈ P◦ w by solving an instance of LPw: max yx, s.t. x ∈ Pw. This can be computed by oracle O and output an optimal solution x∗. If yTx∗ ≤ 1, then y ∈ P◦ w, otherwise yTx∗ = 1 is a hyperplane separating y from P◦ w. Till now, we have constructed a separation oracle for P◦ w using O. Therefore, we can maximize any linear function over P◦ w in polynomial time, again by ellipsoid method. Then, similar arguments as above implies that we can construct a separation oracle for P◦◦ w = {x : yx ≤ 1,∀y ∈ P◦ w} = Pw. This proves that the separation oracle problem for Pw reduces to LPw. Then we have, LPg reduces by ellipsoid method to the separation oracle problem for Pg , which again reduces to the separation oracle problem for Pw (by Lemma 1), which then reduces to LPw.
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