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A Fast Approximation Algorithm for the Subset-Sum Problem

Bartosz Przydatek

The subset-sum problem (SSP) is defined as follows: given a positive integer bound and a set of $n$ positive integers find a subset whose sum is closest to, but not greater than, the bound. We present a randomized approximation algorithm for this problem with linear space complexity and time complexity of $O(n\log n)$. Experiments with random uniformly-distributed instances of SSP show that our algorithm outperforms, both in running time and average error, Martello and Toth's [MartelloToth'84] quadratic greedy search, whose time complexity is $O(n^2)$.

We propose conjectures on the expected error of our algorithm for uniformly-distributed instances of SSP and provide some analytical arguments justifying these conjectures. We present also results of numerous tests.