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ps3sol - 15.083/6.859J Integer Programming Combinatorial...

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15.083/6.859J Integer Programming & Combinatorial Optimization Solutions to Problem Set 3 October 19, 2004 1. BW 4.9 Comparison of relaxations for an assignment problem with a side con- straint If we relax the third constraint, the remaining constraints define an integral polyhedron. Sim- ilarly, if we relax the third constraint and the first set of constraints, the remaining constraints form an integral polyhedron. Hence Z = Z D 1 = Z D 4 . Let X 3 be the polyhedron defined when only the first set of constraints are relaxed, and X 2 be the one defined when the first and the second set of constraints are relaxed. Since X 3 X 2 , Z Z D 2 Z D 3 Z IP . 2. BW 4.10 Let P denote the convex hull of the vectors of start times t = ( t 1 , . . . , t n ) under all possible sequence ordering of jobs and r be the vector of release times. Then, a lower bound for the problem with release times is given by the optimal value of the linear program Z lo = min w j t j j s.t. t j r j j = 1 , . . . , n t P.
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