ps3sol

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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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 deFne an integral polyhedron. Sim- ilarly, if we relax the third constraint and the Frst set of constraints, the remaining constraints form an integral polyhedron. Hence Z = Z D 1 = Z D 4 . Let X 3 be the polyhedron deFned when only the Frst set of constraints are relaxed, and X 2 be the one deFned when the Frst 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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online