11Y16 Algorithms; complexity [See also 68Q25]
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Improved lowdensity subset sum algorithms
(1992)

Matthijs J. Coster
Antoine Joux
Brian A. LaMacchia
Andrew M. Odlyzko
Claus Peter Schnorr
Jacques Stern
 The general subset sum problem is NPcomplete. 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 sufficiently low density. Both methods rely on basis reduction algorithms to find short nonzero vectors in special lattices. The LagariasOdlyzko algorithm would solve almost all subset sum problems of density < 0.6463 . . . in polynomial time if it could invoke a polynomialtime algorithm for finding the shortest nonzero vector in a lattice. This paper presents two modifications of that algorithm, either one of which would solve almost all problems of density < 0.9408 . . . if it could find shortest nonzero vectors in lattices. These modifications also yield dramatic improvements in practice when they are combined with known lattice basis reduction algorithms.