New generic algorithms for hard knapsacks

Abstract

In this paper, we study the complexity of solving hard knapsack problems, i.e., knapsacks with a density close to 1 where lattice-based low density attacks are not an option. For such knapsacks, the current state-of-the-art is a 31-year old algorithm by Schroeppel and Shamir which is based on birthday paradox techniques and yields a running time of for knapsacks of n elements and uses storage. We propose here two new algorithms which improve on this bound, finally lowering the running time down to either or under a reasonable heuristic. We also demonstrate the practicality of these algorithms with an implementation. © 2010 Springer-Verlag.