# knapsack

0th

Percentile

##### 0-1 Knapsack Problem

Solves the 0-1 (binary) single knapsack problem.

Keywords
optimize
##### Usage
knapsack(w, p, cap)
##### Arguments
w

integer vector of weights.

p

integer vector of profits.

cap

maximal capacity of the knapsack, integer too.

##### Details

knapsack solves the 0-1, or: binary, single knapsack problem by using the dynamic programming approach. The problem can be formulated as:

Maximize sum(x*p) such that sum(x*w) <= cap, where x is a vector with x[i] == 0 or 1.

##### Value

A list with components capacity, profit, and indices.

##### Note

Will be replaced by a compiled version.

##### References

Papadimitriou, C. H., and K. Steiglitz (1998). Combinatorial Optimization: Algorithms and Complexity. Dover Publications 1982, 1998.

Horowitz, E., and S. Sahni (1978). Fundamentals of Computer Algorithms. Computer Science Press, Rockville, ML.

knapsack::knapsack

• knapsack
##### Examples
# NOT RUN {
# Example 1
p <- c(15, 100, 90, 60, 40, 15, 10,  1)
w <- c( 2,  20, 20, 30, 40, 30, 60, 10)
cap <- 102
(is <- knapsack(w, p, cap))
#  1 2 3 4 6 , capacity 102 and total profit 280

## Example 2
p <- c(70, 20, 39, 37, 7, 5, 10)
w <- c(31, 10, 20, 19, 4, 3,  6)
cap <- 50
(is <- knapsack(w, p, cap))
#  1 4 , capacity 50 and total profit 107
# }