CS 600 Test 2 Solutions

Fall 2003

1. Draw the feasible region, indicate how the line $z = k$ moves as the constant $k$ increases and graphically solve the following problem:

Maximize $z = 2x_{1} + x_{2}$
Subject to:
$-x_{1} + x_{2} \leq 2$
$x_{1} + 3x_{2} \leq 10$
$3x_{1} + x_{2} \leq 10$
$x_{1} - x_{2} \leq 2$
$x_{1}, x_{2} \geq 0$ [20]









2. For the following optimization problem answer the questions below:

Maximize $5x_{1} + x_{2} + 3x_{3}$
Subject to:
$x_{1} - 2x_{2} - x_{3} \leq -3$
$2x_{1} - 3x_{2} + x_{3} \leq -5$
$x_{1} + x_{2} - 3x_{3} \leq 2$
$x_{1}, x_{2}, x_{3} \geq 0$

(a) Find a basic solution with the slack variables as the basic variables. [7]

$x_{4} = -3 - x_{1} + 2x_{2} + x_{3}$
$x_{5} = -5 - 2x_{1} + 3x_{2} - x_{3}$
$x_{6} = 2 - x_{1} - x_{2} + 3x_{3}$

This is not fasible because $x_{4}$ and $x_{5}$ are negative.

(b) If the basic solution of (a) is not feasible, then find an initial basic feasible solution and the corresponding dictionary for the given optimization problem. If it is feasible, solve the given optimization problem by the simplex method. [23]

Auxiliary Problem:
Max $-x_{0}$ (from Min $x_{0}$)
Subject to:
$x_{1} - 2x_{2} - x_{3} - x_{0} \leq -3$
$2x_{1} - 3x_{2} + x_{3} - x_{0} \leq -5$
$x_{1} + x_{2} - 3x_{3} - x_{0} \leq 2$
$x_{0}, x_{1}, x_{2}, x_{3} \geq 0$

Introducing slack variables we get
$x_{1} - 2x_{2} - x_{3} - x_{0} + x_{4} = -3$
$2x_{1} - 3x_{2} + x_{3} - x_{0} + x_{5} = -5$
$x_{1} + x_{2} - 3x_{3} - x_{0} + x_{6} = 2$
$x_{0}, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6} \geq 0$
$z = -x_{0}$

Switching $x_{0}$ with $x_{5}$, since -5 is the most negative, we get

$x_{0} = 5 + 2x_{1} - 3x_{2} + x_{3} + x_{5}$
$x_{4} = 2 + x_{1} - x_{2} + 2x_{3} + x_{5}$
$x_{6} = 7 + x_{1} - 4x_{2} + 4x_{3} + x_{5}$
$z = -5 - 2x_{1} + 3x_{2} - x_{3} - x_{5}$

Interchanging $x_{0}$ with $x_{2}$, since $x_{2}$ has the largest coefficient, we get

$x_{2} = 5/3 - 1/3 x_{0} + 2/3 x_{1} + 1/3 x_{3} + 1/3 x_{5}$
$x_{4} = 1/3 + 1/3 x_{0} + 1/3 x_{1} + 5/3 x_{3} + 2/3 x_{5}$
$x_{6} = 1/3 + + 4/3 x_{0} - 5/3 x_{1} + 8/3 x_{3} -1/3 x_{5}$
$z = -x_{0}$

Thus the initial feasible basic solution for the original problem is

$x_{1} = x_{3} = x_{5} = 0$, $x_{2} = 5/3, x_{4} = 1/3, x_{6} = 1/3$.

3. Answer whether or not the following statements are true. You DO NOT need to give your reasons. [20]

(a) If a problem is NP-complete, then it can be solved by back tracking a polynomial number of times in the worst case.
(b) An NP-complete problem can be solved in $O(a^{n^{2}})$ time for some constant $a$ in the worst case, where $n$ is the problem size.
(c) When the simplex method is applied to a linear programming problem in canonical form, the value of the objective function always increases.
(d) The simplex method for linear programming is a polynomial time algorithm.

"No" to all.

4. Prove that the following problem is NP-complete:
$k$ frequencies must be assigned to $n$ TV stations. If two TV stations are geographically close, they can not be assigned the same frequency to avoid interference, else they can be given the same frequency. Given the distance for each pair of stations and the minimum distance for assigning the same frequency without interference, find out whether or not $k$ frequencies can be assigned to the $n$ stations so that they do not interfere each other.

You may assume that the following problems are NP-complete:
Bin Packing, Partition, Graph Coloring, Knapsack, 3SAT and 3D Matching. [30]

Transform "Graph Color" to this problem. Do NOT transform this to graph color.