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Knapsack Problem: Greedy vs. Brute Force. pp 313-317 (Section 7.6). Greedy Approach. To solve problems you have to make decisions. At each decision point, you pick the greedy (or best) option. i.e., make an optimal move given what you know For some problems a greed strategy

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greedy approach
Greedy Approach
  • To solve problems you have to make decisions.
  • At each decision point, you pick the greedy (or best) option.
    • i.e., make an optimal move given what you know
  • For some problems a greed strategy
    • produces an optimal solution
    • produces a very bad solution
continuous knapsack problem
Continuous Knapsack Problem
  • Continuous Knapsack Variation
    • Input: N chapters
    • Goal: Maximize importance
    • Constraint: Report must be 600 pages
continuous knapsack problem4
Continuous Knapsack Problem

Step 1: Pick the chapter with the most importance (Chapter 4). Maximize the importance by choosing all 150 pages

continuous knapsack problem5
Continuous Knapsack Problem

Step 2: Greedy choice… pick the most important remaining chapter:Chapter 1 and 2 are tied, pick both, i.e., 270 pages

Report (600 pages max):All of Chapter 4 (importance 8)450 pages leftTotal Importance = 8

continuous knapsack problem6
Continuous Knapsack Problem

Step 3: Greedy choice… pick Chapter 3.However, we can only pick 90% of it, i.e., 180 out of 200 pages.

Report (600 pages max):All of Chapter 4 (importance 8)All of Chapter 1 (importance 5)All of Chapter 2 (importance 5)180 pages leftTotal Importance = 8 + 5 + 5

continuous knapsack problem7
Continuous Knapsack Problem

Is the final answer optimal?

Can you pick page differently and beat 21.6?

Report (600 pages max):All of Chapter 4 (importance 8)All of Chapter 1 (importance 5)All of Chapter 2 (importance 5)90% of Chapter 3 (importance 4)0 pages leftTotal Importance = 8 + 5 + 5 + 4*0.9 = 21.6

constrained optimization problems
Constrained Optimization Problems
  • Many different real-world problem can be reduced to constrained optimization problems.
    • Some factor needs to be maximized or minimized (Goal)
    • Real-world constraints limit the choices you can make (Constraints)
constrained optimization problems9
Constrained Optimization Problems
  • Examples:
    • Package routing
      • Input: N packages and N delivery points
      • Goal: Minimize the package delivery time
      • Constrains: X number of trucks
    • CPU allocation
      • Input: N processing jobs each with a priority
      • Goal: Maximize total priority throughput
      • Constaints: X processor, Y memory, Z time
non continuous knapsack problem
Non-Continuous Knapsack Problem
  • Greedy approach does NOT work.
  • Just call it the Knapsack Problem
    • Constraint: Sack holds 100 lbs.
    • Goal: Maximize sack value
    • Input: Packages
greedy lowest weight first with one backtracking step
Greedy: Lowest Weight Firstwith one Backtracking step

79 lbs $83--------------------------

21 lbs remaining26 lbs $30 item avail. removing the lowest valued item 5 lbs. or more.---------------------------95 lbs $103

greedy highest value first with one restart step
Greedy: Highest Value Firstwith one Restart step

99 lbs $100---------------------Skip the most valuable item and start again---------------------98 lbs $107

greedy highest density with one backtracking step
Greedy: Highest Densitywith one Backtracking step

90 lbs $103------------------10 lbs remaining15 lbs $10 item avail. removing the lowest valued item 5 lbs. or more.---------------------------96 lbs $102