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Unformatted text preview: Massachusetts Institute of Technology Handout 12 6.854J/18.415J: Advanced Algorithms Mon, October 26, 2009 David Karger Problem Set 5 Solutions Mon, October 26, 2009 Problem 1. (a) We know any LP P can be written in standard form, i.e. minimize c T x subject to Ax = b,x 0. Now, consider the dual LP: maximize y T b subject to y T A c . If the primal has an optimal solution, the dual has an optimal solution of the same value. Therefore, consider the following LP Q : minimize 0 subject to Ax = b,x ,y T A c,c T x = y T b . Consider P : it is either feasible with an optimal solution, feasible but unbounded, or infeasible. Clearly, Q will remain infeasible P was infeasible. If there is an optimal solution to Q , we can recover it and produce an optimal solution to P . Finally, we can consider the original LP P without minimization (call it R ): minimize 0 subject to Ax = b,x 0. If R is feasible and Q was infeasible, then P was feasible but unbounded. This allows us to distinguish all three cases. (b) The dual of this linear program is maximize y T b subject to y T A 0. (c) If the primal is feasible, it has as an optimum value 0. Therefore, the optimum value for the dual is 0. So therefore y = 0 is an optimum solution for the dual. (d) Give the algorithm the LP Q described in part (a). Constructing Q takes less than O (( m + n ) k ) time. Now, give as the dual solution to Q the zero vector. By part (a), this algorithm will solve the LP, i.e. it can distinguish the three cases that an LP can fall into. Problem 2. (a) The conservation constraints on f ij ensure that the flow defined by f ij is a circulation. The net flow in the circulation is 1. Notice that there are no capacity constraints on the solution....
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This note was uploaded on 11/24/2009 for the course CS 6.854/18.4 taught by Professor Davidkarger during the Fall '09 term at MIT.
- Fall '09