# Solving Medium-Density Subset Sum Problems in Expected Polynomial Time

## Abraham D. Flaxman and Bartosz Przydatek

```
```The subset sum problem (SSP) (given n numbers and a target bound B,
find a subset of the numbers summing to B), is a classic NP-hard problem.
The hardness of SSP varies greatly with the density of the problem.
In particular, when m, the logarithm of the largest input number,
is at least c*n for some constant c, the problem can be solved by a reduction
to finding a short vector in a lattice. On the other hand, when m = O(log n)
the problem can be solved in polynomial time using dynamic programming or some
other algorithms especially designed for dense instances. However, as far as
we are aware, all known algorithms for dense SSP take at least $\Omega(2^m)$
time, and no polynomial time algorithm is known which solves SSP when
$m = \omega(log n)$ (and m = o(n)).

We present an expected polynomial time algorithm for solving uniformly random
instances of the subset sum problem over the domain $Z_M$, with m=O((log n)^2).
To the best of our knowledge, this is the first algorithm working efficiently
beyond the magnitude bound of O(log n), thus narrowing the interval of
hard-to-solve SSP instances.