The Subset Sum Problem

SUBSET_SUM is a MATLAB program which seeks solutions of the subset sum problem.

The task is to compute a sum S using a selected subset of a given set of N weights.

SUBSET_SUM_NEXT works by backtracking, returning all possible solutions one at a time, keeping track of the selected weights using a 0/1 mask vector of size N.

SUBSET_SUM_TABLE works by a kind of dynamic programming approach, constructing a table of all possible sums from 1 to S. The storage required is N * S, so for large S this can be an issue.

SUBSET_SUM_FIND works by brute force, trying every possible subset to see if it sums to the desired value. It uses the bits of a 32 bit integer to keep track of the possibilities, and hence cannot work with more N = 31 weights.


The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.


SUBSET_SUM is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

CHANGE_MAKING, a MATLAB library which considers the change making problem, in which a given sum is to be formed using coins of various denominations.

COMBINATION_LOCK, a MATLAB program which simulates the process of determining the secret combination of a lock.

COMBO, a MATLAB library which includes many combinatorial routines.

PARTITION_PROBLEM, a MATLAB library which seeks solutions of the partition problem, splitting a set of integers into two subsets with equal sum.

SATISFY, a MATLAB program which demonstrates, for a particular circuit, an exhaustive search for solutions of the circuit satisfiability problem.

SUBSET_SUM, a dataset directory which contains examples of the subset sum problem, in which a set of numbers is given, and is desired to find at least one subset that sums to a given target value.

SUBSET_SUM_TASKS, a MATLAB program which runs in parallel, seeking solutions of the "subset sum" version of the knapsack problem. Instructions are available on how to run the job, via MATLAB's BATCH facility, on a remote system such as Virginia Tech's ITHACA cluster.

TSP_BRUTE, a MATLAB program which reads a file of city-to-city distances and solves the traveling salesperson problem, using brute force.


  1. Alexander Dewdney,
    The Turing Omnibus,
    Freeman, 1989,
    ISBN13: 9780716781547,
    LC: QA76.D45.
  2. Donald Kreher, Douglas Simpson,
    Combinatorial Algorithms,
    CRC Press, 1998,
    ISBN: 0-8493-3988-X,
    LC: QA164.K73.
  3. Silvano Martello, Paolo Toth,
    Knapsack Problems: Algorithms and Computer Implementations,
    Wiley, 1990,
    ISBN: 0-471-92420-2,
    LC: QA267.7.M37.

Source Code:

Examples and Tests:

You can go up one level to the MATLAB source codes.

Last modified on 14 July 2017.