Knapsack (2)

Brute Force Approaches

• Techniques for finding optimal solutions to hard problems “relatively” quickly

• Backtracking

• Branch and Bound

• Combining efficient solutions with brute force approaches

• Other issues

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

• and the sum of values of items in S is maximized

Greedy Does Not Work

• What are possible greedy strategies?– Highest Density First– Highest Value First– Lowest Weight First

• Prove these are not optimal for the 0-1 knapsack problem.

Multi-constraint Knapsack

• Input– Capacity K1 and K2

– n items with integer weights wi, size si, and values vi

• Goal– Output a set of items S such that

• the sum of weights of items in S is at most K1

• the sum of sizes of items in S is at most K2

• and the sum of values of items in S is maximized

• Can we use dynamic programming?

Situation where dynamic programming does not work

What if our problem can’t be described with integers?

wA = 2 vA = $40

wB = vB = $50

wC = 1.98 vC = $100

wD = 5 vD = $95

wE = 3 vE = $30

We have to resort to brute force….

Brute Force

• Generate all possible solutions– With n items, there are 2n solutions to be

generated – Check each to see if they satisfy the constraint– Save maximum solution that satisfies constraint

• Can be represented as a tree

Brute Force: BranchingA

B

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In Out

Weight = 15.12Value = $315



Backtracking

• In the tree representation, we can think of the previous algorithm doing a DFS in the tree

• If we reach a point where a solution no longer is feasible, there is no need to continue exploring

• We can “backtrack” from this point and potentially cut off much of the tree and many of the solutions

• In the given example, backtracking would be much more effective if we had even more items or a smaller knapsack capacity

BacktrackingA

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In Out



2, $40

, $50

1.98, $100

5, $95

3, $30

Branch and Bound

• We can backtrack if we know the best possible solution in current subtree is worse than current best solution obtained so far.

• Estimates for improvement given current ordering– A down can give $315– B down -> $275– C down -> $225– D down -> $125– E down -> $30

Branch and BoundA

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In Out



2, $40

, $50

1.98, $100

5, $95

3, $30

Generating a good bound

• The key to branch and bound is having a good initial bound.

• How might we generate a good initial bound?

Order of items

• Since there is no fixed ordering of items, is there a better way to order the input items?– Highest weight first

• Generate infeasible solutions quickly– Highest density first

• Generate good solution quickly– Which is better depends on input

2, $40

, $50

1.98, $100

5, $95

3, $30

Heaviest on TopD

B

E

A A

C C

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In Out





(Max remaining = $220)

D: 5, $95

B: , $50

E: 3, $30

A: 2, $40

C: 1.98, $100

Best Density on TopC

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In Out


C: 1.98, $100 A: 2, $40

D: 5, $95

B: , $50

E: 3, $30

Greedy and Brute Force

• Suppose half the inputs to our knapsack problem all have the same weight.– If all inputs had the same weight, we could

implement a greedy solution– However, since half do not, we cannot use

greedy alone to find optimal solution

• Combine brute-force approach with greedy to find optimal solution more quickly.

The Breakdown...F1

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F4 F4

F2

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In Out

Greedyon half!

F5 F5 F5 F5 F5 F5 F5 F5 F5

Algorithm

• Backtrack on half the inputs

• At leafs, apply greedy strategy on the other half of the inputs

• Comparison– Pure brute force: O(2n)– Combination: O(n2n/2)

How much better?

• Assumptions– Suppose n=50 – We can test 1,000,000 solutions/second.

• 250 would take over 35 years

• 225 can be generated in half an hour– Plus marginal time to generate greedy solution

for each of the 225 solutions

Another combination

• Suppose half the inputs to our knapsack problem have small integral weights/values while the other half have real weights/values.

• How can we combine approaches to solve this efficiently?

The n-Queens Problem

• Input– Positive integer n

• Task– Place n queens on an n by n chessboard so that

no two queens attack each other (on same row, column, diagonal), or report that this is impossible

• Solve the n-queens problem for n = 1, 2, 3

n=4

• Pure brute force search– 16 squares on a 4 by 4 chessboard– Leads to 16 * 15 * 14 * 13 = 43,680 possible

solutions

• Improvements– At most one queen per row: 44 = 256 possible

solutions– Backtracking: If two queens already attack

before final queen placed, backtrack

Larger values of n

• n=8– At most one queen per row: 88 = 16,777,216– Early backtracking: 2057 nodes total– Time to find first solution: 114 nodes

• n=12– At most one queen per row: 1212 – Early backtracking: 856,189 nodes total– Time to find first solution: 262 nodes

The ProblemIn the U.S. navy, the SEALS are each specially trained in a wide variety of skills so that small teams can handle a multitude of missions.

If there are k different skills needed for a mission, and n SEAL members that can be assigned to the team, find the smallest team that will cover all of the required skills.

Andersen knows hand-to-hand, first aid, and camouflageButler knows hand-to-hand and snaresCunningham knows hand-to-handDouglas knows hand-to-hand, sniping, diplomacy, and snaresEckers knows first-aid, sniping, and diplomacy

Greedy Algorithm

• What is the obvious greedy algorithm for this problem?

• Find a counter-example to the optimality of greedy for this problem.

Brute Force Approach

• What is the brute-force approach?

• How can we simplify the problem as far as possible in polynomial time?

• How can we set bounds on the solutions?

• When will we need to do backtracking?

• What order should we test the potential members in when branching?

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Knapsack (2)