Improved Low - Density Subset

The general subset sum problem is NP-complete. However, there are two algorithms, one due to Brickell and the other to Lagarias and Odlyzko, which in polynomial time solve almost all subset sum problems of suuciently low density. Both methods rely on basis reduction algorithms to nd short non-zero vectors in special lattices. The Lagarias-Odlyzko algorithm would solve almost all subset sum problems of density < 0:6463 : : : in polynomial time if it could invoke a polynomial-time algorithm for nding the shortest non-zero vector in a lattice. This paper presents two modiications of that algorithm, either one of which would solve almost all problems of density < 0:9408 : : : if it could nd shortest non-zero vectors in lattices. These modiications also yield dramatic improvements in practice when they are combined with known lattice basis reduction algorithms.