Solving Medium-Density Subset Sum Problems in Expected Polynomial Time

Abstract

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 $m=\mathcal{O}(log n)$ 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 Ω(2m) time, and no polynomial time algorithm is known which solves SSP when m = ω(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 ℤM, with $m=\mathcal{O}((log n)^{2})$. To the best of our knowledge, this is the first algorithm working efficiently beyond the magnitude bound of $\mathcal{O}(log n)$, thus narrowing the interval of hard-to-solve SSP instances.