Solving Medium-Density Subset Sum Problems in Expected Polynomial Time

The subset sum problem (SSP) (given n numbers and a target bound B, find a subset of the numbers summing to B), is one of the classical NP-hard problems. The hardness of SSP varies greatly with the density of the problem. In particular, if 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 such 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 with significantly lower density, i.e., when m = ω(log n) (and m = o(n)). We present an expected polynomial time algorithm for solving uniformly random instances of subset sum problem over the domain Z2m , with m = O((log n)2 / log log n). To the best of...

• Type: Conference
• Year: 2005
• Where: STACS
• Authors: Abraham Flaxman, Bartosz Przydatek
• Keywords: Polynomial Time, Polynomial Time Algorithm, STACS 2005, Subset Sum Problem, Theoretical Computer Science