The Knapsack Problem

1
The Knapsack Problem

• Input
• Capacity K
• n items with weights wi and values vi
• Goal
• Output a set of items S such that
• the sum of weights of items in S is at most K
• the sum of values of items in S is maximized

2
Variations

• Fractional
• You may take a fraction of the entire item
• All or nothing
• You must take all of an item or none of it
• Special cases of All or nothing
• All items have same weight (or same value)
• All items have same density (value/weight)
• For which of the above variations does a greedy
  strategy work?
• For which of the above variations does a greedy
  strategy NOT work?

3
Dynamic Programming

• Consider the all or nothing version
• Assume that all weights/values are reasonably
  sized integers
• We can use dynamic programming to solve this
  problem (even though there is no explicit
  ordering among the items)

4
Definining subproblems

• Define P(i,w) to be the problem of choosing a set
  of objects from the first i objects that
  maximizes value subject to weight constraint of
  w.
• Impose an arbitrary ordering on the items
• V(i,w) is the value of this set of items
• Original problem corresponds to V(n, K)

5
Recurrence Relation/Running Time

• V(i,w) max (V(i-1,w-wi) vi, V(i-1, w))
• A maximal solution for P(i,w) either
• uses item i (first term in max)
• or does NOT use item i (second term in max)
• V(0,w) 0 (no items to choose from)
• V(i,0) 0 (no weight allowed)
• What is the running time of this solution?
• Number of table entries
• Time to fill each entry

6
Example
Items
wA 2 vA 40 wB 3 vB 50 wC 1 vC
100 wD 5 vD 95 wE 3 vE 30
Weight
7
Define Subproblems Based on Value

• Define P(i,w) to be the problem of choosing a set
  of objects from the first i objects that
  maximizes value subject to weight constraint of
  w.
• V(i,w) is the value of this set of items
• Original problem corresponds to V(n, W)

• Define P(i,v) to be