Greedy algorithms
1. Consider the set of items S={a, b, c, d, e, f, g, h}where the benefits
and weights are given in the table below. Note that I have computed
the value index of each item (benefit/weight) and that is the last col-
umn in the table.
item benefit weight vi
a 14 3 42
3
b 5 1 5
c 10 6 1 2
3
d 12 4 3
e 8 2 4
f 10 4 21
2
g 16 8 2
h 9 9 1
Solve the Fractional Knapsack Problem for this set of items, where the
maximum total allowed weight is Wmax = 15.
Using the value indices, we select the items in order b,a, e, d, f , g, c, h
until the knapsack is full.
Then we
Pick item b. Total benefit (so far) = 5. Total weight (so far) = 1.
Pick item a. total benefit = 19. Total weight = 4.
Pick item e. Total benefit = 27. Total weight = 6.
Pick item d. Total benefit = 39. Total weight = 10.
Pick item f. Total benefit = 49. Total weight = 14.
Pick 1
8of item g. Total benefit = 49 + 1
8(16) = 51. Total weight =
14 + 1
8(8) = 15.
Therefore, our solution is to take items b, a, e, d, f, 1
8gfor a total benefit
of 51.
2. Let T={(1,3),(2,4),(3,5),(2,7),(4,6),(5,6),(3,7),(5,8),(6,10),(7,9),(8,10)}
denote the (start,finish) times for a collection of 11 tasks.
(a) Solve the Interval Scheduling Problem for this collection of tasks
(i.e. find the maximum number of tasks that can be scheduled on
a single machine, and give a set of compatible tasks that achieves
this maximum).
Solution: For the Interval Scheduling Problem, we first sort the
intervals by their finish times.
Note that I have numbered the tasks in order of their finish time.
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