Lecture 1 #

Computational Models #

An objective function that is to be maximized or minimized, e.g.,
- Minimize time spent traveling from New York to Boston
A set of constraints (possibly empty) that must be honored, e.g.,
- Cannot spend more than $100
- Must be in Boston before 5:00PM

You have limited strength, so there is a maximum weight knapsack that you can carry
You would like to take more stuff than you can carry
How do you choose which stuff to take and which to leave behind?
Two variants
- 0/1 knapsack problem
- Continuous or fractional knapsack problem

Each item is represented by a pair, <value, weight>
The knapsack can accommodate items with a total weight of no more than w
A vector, L, of length n, represents the set of available items. Each element of the vector is an item
A vector, V, of length n, is used to indicate whether or not items are taken. If V[i] = 1, item I[i] is taken. If V[i] = 0, item I[i] is not taken
Find a V that Maximizes:
- $\displaystyle\sum_{i=0}^{n-1}V[i]*I[i].\text{value}$
- Subject to the constraint that:
  - $\displaystyle\sum_{i=0}^{n-1}V[i]*I[i].\text{weight} \le w$

Procedure:
- 1. Enumerate all possible combinations of items. That is to say, generate all subsets of the set of subjects. This is called the power set.
- 1. Remove all of the combinations whose total units exceeds the allowed weight.
- 1. From the remaining combinations choose any one whose value is the largest.
Dark Side:
- Will take a considerable time to get the power set.
- There will lots of Vs to indicate whether or not items are taken.

Put “best” available item in knapsack
Procedure:
- Define what the best means.
- Sort the items.
- Take the items from the best to the worst, stop when it hit the maximum.
Dark Side:
- Sequence of locally “optimal” choices don’t always yield a globally optimal solution