Subset-Sum Problem
Exact algorithm - Dynamic programming

PROBLEM
In the subset-sum problem we wish to find a subset of A.1,...,A.N whose sum is as large as possible but not larger than T (capacity of the knapsack).

ALGORITHM
The time complexity of the following exact algorihm is O(MIN(2**(N+1),N*T). The algorithm encodes the result as bit string. For example:

 N=10; T = 69 A.1 = 5; A.2 = 9; A.3 = 13; A.4 = 4; A.5 = 1 A.6 = 17; A.7 = 11; A.8 = 2; A.9 = 3; A.10 = 12

The result is 743, it is the bit string 1011100111 and it means that elements with indexes (read the bit string from the right to the left) 1, 2, 3, 6, 7, 8, 10 belong to subset. Proof: A.1 (5) + A.2 (9) + A.3 (13) + A.6 (17) + A.7 (11) + A.8 (2) + A.10 (12) = 69

IMPLEMENTATION
Unit: internal subroutine

Global variables: array A.1,...,A.N of positive integers, output array X.

Parameters: a positive integer N, a positive integer T

Result: output array X., where X.J=1 if item A.J is selected; or X.J=0 otherwise (for J=1,...,N)

 DPS: procedure expose A. X. parse arg N, T /* for N <= 10000 */ numeric digits 3011; Nd = 2 ** N - 1 numeric digits LENGTH(Nd) A1.0 = 0; X.0 = 0; S = 1 B = 2; A1.1 = A.1; X.1 = 1; M = 2 do until M > N | A1.S = T   I = 0; K = 0; H = 1   Y = A.M; Sp1 = S + 1; A1.Sp1 = 1E+100   A2.0 = 0; X2.0 = 0   do while MIN(Y, A1.H) <= T     K = K + 1     if A1.H <= Y       then do         A2.K = A1.H; X2.K = X.H; H = H + 1       end       else do         A2.K = Y; X2.K = X.I + B       end     if A2.K = Y       then do         I = I + 1; Y = A1.I + A.M       end   end   S = K; B = 2 * B   do J = 1 to S     A1.J = A2.J; X.J = X2.J   end   M = M + 1 end BinXS = X.S do J = 1 while BinXS > 0   X.J = BinXS // 2   BinXS = BinXS % 2 end do J = J to N; X.J = 0; end return

COMPARISON

For N=100;T=25557 and the array A. created by statements:

 Seed = RANDOM(1, 1, 481989) do J = 1 to N   A.J = RANDOM(1, 1000) end

I compared the algorithms for solution of the Subset-sum problem and my algorithm DIOPHANT for solution of the diofantine equations.

Notes:
I halted the EXACT_SUBSET_SUM after 30 minutes of computations. For APPROX_SUBSET_SUM I used the value Epsilon=0.5

Subset-sum problem - Comparison of Algorithms
Algorithm Subset sum Seconds
GS 25554      0.05
DPS 25557   240.24
APPROX_SUBSET_SUM 25436    12.31
DIOPHANT 25557      0.82

CONNECTIONS

Literature
Martello S., Toth P. Knapsack Problems: Algorithms nad Computer Implementations
Chichester, John Wiley & sons 1990